diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/_arf_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/_arf_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..9bbd5682615880cafb58547a8b2cce5c4389baf4 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/_arf_generate.py @@ -0,0 +1,23 @@ +import pickle +import pandas as pd + +n_target = int(7636) +with open("/work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf_model.pkl", "rb") as f: + model = pickle.load(f) +syn = model.forge(n=n_target) +syn = syn.reset_index(drop=True) +if len(syn) > n_target: + syn = syn.iloc[:n_target] +elif len(syn) < n_target: + parts = [syn] + tries = 0 + while sum(len(p) for p in parts) < n_target and tries < 64: + tries += 1 + need = n_target - sum(len(p) for p in parts) + chunk = model.forge(n=max(need, 1)).reset_index(drop=True) + if len(chunk) == 0: + break + parts.append(chunk) + syn = pd.concat(parts, ignore_index=True).iloc[:n_target] +syn.to_csv("/work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf-c6-7636-20260422_060120.csv", index=False) +print(f"[ARF] Generated {len(syn)} rows (requested {n_target}) -> /work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf-c6-7636-20260422_060120.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/_arf_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/_arf_train.py new file mode 100644 index 0000000000000000000000000000000000000000..790fc9c830a5eb9ed79bba3713678483381ce874 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/_arf_train.py @@ -0,0 +1,37 @@ +import pickle +import numpy as np +import pandas as pd +from arfpy import arf + +def _sanitize_for_arf(df: pd.DataFrame) -> pd.DataFrame: + """缓解 forge 阶段 scipy.stats.truncnorm / 除零：处理 inf、NaN 与极端尾部。""" + df = df.replace([np.inf, -np.inf], np.nan) + df = df.dropna(axis=1, how="all") + for col in df.select_dtypes(include=[np.number]).columns: + med = df[col].median() + if pd.isna(med): + med = 0.0 + df[col] = df[col].fillna(med) + nu = int(df[col].nunique(dropna=True)) + if nu <= 1: + continue + lo, hi = df[col].quantile(0.001), df[col].quantile(0.999) + if pd.notna(lo) and pd.notna(hi) and lo < hi: + df[col] = df[col].clip(lo, hi) + return df + +df = pd.read_csv("/work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/train.csv") +df = _sanitize_for_arf(df) +print(f"[ARF] Training on {len(df)} rows, {len(df.columns)} cols") + +model = arf.arf(x=df) +if hasattr(model, "fit"): + model.fit() +elif hasattr(model, "forde"): + model.forde() +else: + raise RuntimeError("arfpy API: no fit() / forde()") + +with open("/work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf_model.pkl", "wb") as f: + pickle.dump(model, f) +print(f"[ARF] Model saved -> /work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf_model.pkl") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/gen_20260422_060120.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/gen_20260422_060120.log new file mode 100644 index 0000000000000000000000000000000000000000..96156dfe7d308bc654b1b74eefee62cfdf233b43 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/gen_20260422_060120.log @@ -0,0 +1,11 @@ +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +[ARF] Generated 7636 rows (requested 7636) -> /work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf-c6-7636-20260422_060120.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..165b543a77e5b9a8f79f31fdf96a244bad6f72bf --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + "model": "arf", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "exists": true, + "size": 849500, + "sha256": "7d8f85a52de0e63e292778c26cb06223383b366c589d4226c3de68b111ba5272" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "exists": true, + "size": 108137, + "sha256": "9ede9f1e2036e743d822e8ed8d7b5e1050159e8fc7b402b758a294f7a14528fe" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv", + "exists": true, + "size": 107696, + "sha256": "d28b60b361526450f0c203ddf50498854cb66ad5c1978516a99c265f529f8e4f" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_profile.json", + "exists": true, + "size": 4145, + "sha256": "70c4d3f4f544b9bff7543f502136d9b1403d8589ad5ef0a9695842d8ef9d5185" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_contract_v1.json", + "exists": true, + "size": 4740, + "sha256": "602750e8159221cf97836d44d530098411b5f2cd6fc47c06776171da79d06593" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..62de672c6242a8fe33594746bc19ea183e203ce7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,169 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "columns": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/public_gate_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68d5e4a7e5533d3434a96c463bb1367c9d25256e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "c6", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "Type of Answer", + "task_type": "classification", + "input_splits": { + "train": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "val": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "test": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/staged_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/staged_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..2d91e2ad8567b11f0b5e9f678080f8c309c4d414 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/staged_input_manifest.json @@ -0,0 +1,174 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/public_gate/public_gate_report.json", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..af87f21066af8eca2e62d6ca02a50b76aa9d9cc6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/runtime_result.json @@ -0,0 +1,15 @@ +{ + "dataset_id": "c6", + "model": "arf", + "run_id": "arf-c6-20260422_055912", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "success", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf-c6-7636-20260422_060120.csv", + "model_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf_model.pkl" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/adapter_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/adapter_report.json new file mode 100644 index 0000000000000000000000000000000000000000..26428e5de6d3f2de0d734632d70ffc0c1de67887 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/adapter_report.json @@ -0,0 +1,7 @@ +{ + "adapter_ready_status": "pass", + "adapter_fail_reason_code": null, + "adapter_fail_detail": null, + "adapter_transforms_applied": [], + "model_input_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/model_input_manifest.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/adapter_transforms_applied.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/adapter_transforms_applied.json new file mode 100644 index 0000000000000000000000000000000000000000..0637a088a01e8ddab3bf3fa98dbe804cbde1a0dc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/adapter_transforms_applied.json @@ -0,0 +1 @@ +[] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/model_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..98cecf6af6fc941deee9c3ebcd2ab2fdbd542cbc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/arf/model_input_manifest.json @@ -0,0 +1,176 @@ +{ + "dataset_id": "c6", + "model": "arf", + "target_column": "Type of Answer", + "task_type": "classification", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + 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system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,542,1,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +128,Lithuania,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +552,Portugal,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +979,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +982,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1420,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +378,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +885,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +899,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1219,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation 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Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1363,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" 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application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +633,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1334,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +483,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1109,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +751,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +988,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +898,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" 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Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian 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combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace 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combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +468,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1331,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +600,Italy,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/train_20260422_055912.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/train_20260422_055912.log new file mode 100644 index 0000000000000000000000000000000000000000..4891293414251f17a5b00133e7e1082aa382773e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/train_20260422_055912.log @@ -0,0 +1,5 @@ +[ARF] Training on 7636 rows, 8 cols +Initial accuracy is 0.9684389732844421 +Iteration number 1 reached accuracy of 0.551925091671032. +Iteration number 2 reached accuracy of 0.46640911471974855. +[ARF] Model saved -> /work/output-SpecializedModels/c6/arf/arf-c6-20260422_055912/arf_model.pkl diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/_bayesnet_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/_bayesnet_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..f8423a54fd6d6e70b3da2f898801502cfc68134e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/_bayesnet_generate.py @@ -0,0 +1,104 @@ + +import pickle +import subprocess +import sys +import warnings + +import numpy as np +import pandas as pd +from pgmpy.sampling import BayesianModelSampling + +warnings.filterwarnings("ignore", category=FutureWarning) + +def _ensure_cloudpickle(): + try: + import cloudpickle # noqa: F401 + except ModuleNotFoundError: + subprocess.check_call( + [sys.executable, "-m", "pip", "install", "--quiet", "cloudpickle"], + ) + +_ensure_cloudpickle() + +with open("/work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_model.pkl", "rb") as f: + bundle = pickle.load(f) + +network = bundle["network"] +inverse = bundle["inverse"] +cols = bundle["column_order"] +integer_columns = set(bundle.get("integer_columns") or []) +full_order = bundle.get("full_column_order") or cols +const_cols = bundle.get("const_cols") or {} + +num_rows = int(7636) +sampler = BayesianModelSampling(network) +raw = sampler.forward_sample(size=num_rows, show_progress=False) +raw = raw.reset_index(drop=True) +if len(raw) > num_rows: + raw = raw.iloc[:num_rows] +_tries = 0 +while len(raw) < num_rows and _tries < 64: + _tries += 1 + nextra = min(10000, num_rows - len(raw)) + more = sampler.forward_sample(size=max(nextra, 1), show_progress=False) + more = more.reset_index(drop=True) + if len(more) == 0: + break + raw = pd.concat([raw, more], ignore_index=True) + if len(raw) > num_rows: + raw = raw.iloc[:num_rows] + +out = pd.DataFrame(index=raw.index) +rng = np.random.default_rng() + +for c in cols: + if c in inverse["categorical"]: + levels = inverse["categorical"][c] + idx = raw[c].astype(int).to_numpy() + idx = np.clip(idx, 0, max(0, len(levels) - 1)) + out[c] = [levels[i] for i in idx] + else: + edges = np.asarray(inverse["continuous"][c], dtype=float) + if edges.size < 2: + out[c] = 0.0 + else: + nbin = edges.size - 1 + res = [] + for k in raw[c].astype(int).to_numpy(): + k = int(k) + if k < 0: + k = 0 + if k >= nbin: + k = nbin - 1 + lo, hi = float(edges[k]), float(edges[k + 1]) + if hi < lo: + lo, hi = hi, lo + v = rng.uniform(lo, hi) + if c in integer_columns: + v = int(round(v)) + res.append(v) + out[c] = res + +final = pd.DataFrame(index=out.index) +for c in full_order: + if c in const_cols: + final[c] = const_cols[c] + elif c in out.columns: + final[c] = out[c] + +dtypes = bundle.get("original_dtypes") or {} +for c, dts in dtypes.items(): + if c not in final.columns: + continue + try: + if "int" in dts: + final[c] = pd.to_numeric(final[c], errors="coerce").astype("Int64") + elif "float" in dts: + final[c] = pd.to_numeric(final[c], errors="coerce") + except Exception: + pass + +if len(final) != num_rows: + final = final.iloc[:num_rows].copy() +final.to_csv("/work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet-c6-7636-20260422_060304.csv", index=False) +print(f"[BayesNet] Generated {len(final)} rows (requested {num_rows}) -> /work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet-c6-7636-20260422_060304.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/_bayesnet_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/_bayesnet_train.py new file mode 100644 index 0000000000000000000000000000000000000000..d6c6895bc60aac0a0bdd26965b6d955ae772e3f7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/_bayesnet_train.py @@ -0,0 +1,118 @@ + +import json +import pickle +import subprocess +import sys +import warnings + +import numpy as np +import pandas as pd +from pgmpy.estimators import TreeSearch +from pgmpy.models import DiscreteBayesianNetwork +warnings.filterwarnings("ignore", category=FutureWarning) + +def _ensure_cloudpickle(): + try: + import cloudpickle # noqa: F401 + except ModuleNotFoundError: + subprocess.check_call( + [sys.executable, "-m", "pip", "install", "--quiet", "cloudpickle"], + ) + +_ensure_cloudpickle() + +with open("/work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_coltypes.json", "r", encoding="utf-8") as _f: + colmeta = json.load(_f) +integer_columns = set(colmeta.get("integer_columns") or []) + +df = pd.read_csv("/work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/public/train.csv") +df = df.dropna(axis=1, how="all") +full_column_order = list(df.columns) + +const_cols = {} +for col in list(df.columns): + if df[col].nunique(dropna=True) <= 1: + const_cols[col] = df[col].iloc[0] if len(df) > 0 else None + df = df.drop(columns=[col]) + print(f"[BayesNet] Dropped zero-variance column '{col}'") + +const_path = "/work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_model.pkl".replace("bayesnet_model.pkl", "const_cols.json") +with open(const_path, "w", encoding="utf-8") as _f: + json.dump({k: str(v) for k, v in const_cols.items()}, _f) + +inverse = {"categorical": {}, "continuous": {}} +enc = pd.DataFrame(index=df.index) +_n_samples = len(df) +_n_plan = sum( + 1 for e in colmeta["columns"] if str(e.get("name", "")) in df.columns +) +max_bins = 10 +if _n_plan > 35 or _n_samples > 200000: + max_bins = 5 +if _n_plan > 55: + max_bins = 4 +print(f"[BayesNet] max_bins={max_bins} (cols_in_df={_n_plan}, rows={_n_samples})") + +for entry in colmeta["columns"]: + name = entry["name"] + if name not in df.columns: + continue + kind = entry["type"] + s = df[name] + if kind == "categorical": + uniques = sorted(s.dropna().unique(), key=lambda x: str(x)) + mapping = {str(v): i for i, v in enumerate(uniques)} + inverse["categorical"][name] = [uniques[i] for i in range(len(uniques))] + enc[name] = s.map(lambda x, m=mapping: m.get(str(x), 0)).astype(int) + else: + s_num = pd.to_numeric(s, errors="coerce") + nu = int(s_num.nunique(dropna=True)) + q = min(max_bins, max(2, nu)) + if nu < 2: + enc[name] = np.zeros(len(s_num), dtype=int) + lo, hi = float(s_num.min()), float(s_num.max()) + inverse["continuous"][name] = [lo, hi] + else: + try: + _, bins = pd.qcut( + s_num, q=q, retbins=True, duplicates="drop" + ) + except Exception: + med = float(s_num.median()) + s2 = s_num.fillna(med) + _, bins = pd.qcut( + s2, q=min(q, 3), retbins=True, duplicates="drop" + ) + bins = np.asarray(bins, dtype=float) + lab = pd.cut( + s_num, bins=bins, labels=False, include_lowest=True + ) + enc[name] = lab.fillna(0).astype(int) + inverse["continuous"][name] = bins.tolist() + +print(f"[BayesNet] Training on {len(enc)} rows, {len(enc.columns)} cols (encoded)") + +enc_struct = enc +if len(enc) > 25000: + enc_struct = enc.sample(n=25000, random_state=0, replace=False) + print(f"[BayesNet] TreeSearch on {len(enc_struct)} rows (subsample; full n={len(enc)})") +dag = TreeSearch(enc_struct).estimate(show_progress=False) +for col in enc.columns: + if col not in dag.nodes(): + dag.add_node(col) + print(f"[BayesNet] Added isolated node to DAG: {col}") +network = DiscreteBayesianNetwork(dag) +network.fit(enc) + +bundle = { + "network": network, + "inverse": inverse, + "column_order": list(enc.columns), + "full_column_order": full_column_order, + "integer_columns": list(integer_columns), + "original_dtypes": {c: str(df[c].dtype) for c in enc.columns}, + "const_cols": const_cols, +} +with open("/work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_model.pkl", "wb") as _f: + pickle.dump(bundle, _f) +print(f"[BayesNet] Model saved -> /work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_model.pkl") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_coltypes.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_coltypes.json new file mode 100644 index 0000000000000000000000000000000000000000..1b5e9b2571099fabb160b233af3b2c15f6b9fd97 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_coltypes.json @@ -0,0 +1,37 @@ +{ + "columns": [ + { + "name": "Student ID", + "type": "continuous" + }, + { + "name": "Student Country", + "type": "categorical" + }, + { + "name": "Question ID", + "type": "continuous" + }, + { + "name": "Type of Answer", + "type": "categorical" + }, + { + "name": "Question Level", + "type": "categorical" + }, + { + "name": "Topic", + "type": "categorical" + }, + { + "name": "Subtopic", + "type": "categorical" + }, + { + "name": "Keywords", + "type": "categorical" + } + ], + "integer_columns": [] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/gen_20260422_060304.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/gen_20260422_060304.log new file mode 100644 index 0000000000000000000000000000000000000000..045d22fc9b7bafeba7bd18f141f80e476c0db51e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/gen_20260422_060304.log @@ -0,0 +1,48 @@ + +========== +== CUDA == +========== + +CUDA Version 12.8.1 + +Container image Copyright (c) 2016-2023, NVIDIA CORPORATION & AFFILIATES. All rights reserved. + +This container image and its contents are governed by the NVIDIA Deep Learning Container License. +By pulling and using the container, you accept the terms and conditions of this license: +https://developer.nvidia.com/ngc/nvidia-deep-learning-container-license + +A copy of this license is made available in this container at /NGC-DL-CONTAINER-LICENSE for your convenience. + +WARNING: The NVIDIA Driver was not detected. GPU functionality will not be available. + Use the NVIDIA Container Toolkit to start this container with GPU support; see + https://docs.nvidia.com/datacenter/cloud-native/ . + +/usr/local/lib/python3.10/dist-packages/pgmpy/estimators/__init__.py:4: FutureWarning: `pgmpy.estimators.StructureScore` is deprecated and will be removed in a future release. Use `pgmpy.structure_score` instead. + from .StructureScore import ( +ERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts. +synthcity 0.2.12 requires arfpy, which is not installed. +synthcity 0.2.12 requires be-great>=0.0.5; python_version >= "3.9", which is not installed. +synthcity 0.2.12 requires decaf-synthetic-data>=0.1.6, which is not installed. +synthcity 0.2.12 requires fastai<2.8, which is not installed. +synthcity 0.2.12 requires fastcore<1.8, which is not installed. +synthcity 0.2.12 requires fflows, which is not installed. +synthcity 0.2.12 requires geomloss, which is not installed. +synthcity 0.2.12 requires importlib-metadata, which is not installed. +synthcity 0.2.12 requires lifelines<0.30.0,>=0.29.0, which is not installed. +synthcity 0.2.12 requires monai, which is not installed. +synthcity 0.2.12 requires nflows>=0.14, which is not installed. +synthcity 0.2.12 requires opacus>=1.3, which is not installed. +synthcity 0.2.12 requires pycox, which is not installed. +synthcity 0.2.12 requires pykeops, which is not installed. +synthcity 0.2.12 requires redis, which is not installed. +synthcity 0.2.12 requires shap, which is not installed. +synthcity 0.2.12 requires tenacity, which is not installed. +synthcity 0.2.12 requires tsai; python_version > "3.7", which is not installed. +synthcity 0.2.12 requires xgbse>=0.3.1, which is not installed. +synthcity 0.2.12 requires networkx<3.0,>2.0, but you have networkx 3.4.2 which is incompatible. +synthcity 0.2.12 requires numpy<2.0,>=1.20, but you have numpy 2.2.6 which is incompatible. +synthcity 0.2.12 requires pgmpy<1.0, but you have pgmpy 1.1.0 which is incompatible. +synthcity 0.2.12 requires torch<2.3,>=2.1, but you have torch 2.8.0+cu128 which is incompatible. +synthcity 0.2.12 requires xgboost<3.0.0, but you have xgboost 3.2.0 which is incompatible. +WARNING: Running pip as the 'root' user can result in broken permissions and conflicting behaviour with the system package manager, possibly rendering your system unusable. It is recommended to use a virtual environment instead: https://pip.pypa.io/warnings/venv. Use the --root-user-action option if you know what you are doing and want to suppress this warning. +[BayesNet] Generated 7636 rows (requested 7636) -> /work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet-c6-7636-20260422_060304.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..756f09e04e5665267d43feb35e42f0e1eea68be6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + "model": "bayesnet", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "exists": true, + "size": 849500, + "sha256": "7d8f85a52de0e63e292778c26cb06223383b366c589d4226c3de68b111ba5272" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "exists": true, + "size": 108137, + "sha256": "9ede9f1e2036e743d822e8ed8d7b5e1050159e8fc7b402b758a294f7a14528fe" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv", + "exists": true, + "size": 107696, + "sha256": "d28b60b361526450f0c203ddf50498854cb66ad5c1978516a99c265f529f8e4f" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_profile.json", + "exists": true, + "size": 4145, + "sha256": "70c4d3f4f544b9bff7543f502136d9b1403d8589ad5ef0a9695842d8ef9d5185" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_contract_v1.json", + "exists": true, + "size": 4740, + "sha256": "602750e8159221cf97836d44d530098411b5f2cd6fc47c06776171da79d06593" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..62de672c6242a8fe33594746bc19ea183e203ce7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,169 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "columns": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/public_gate_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68d5e4a7e5533d3434a96c463bb1367c9d25256e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "c6", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "Type of Answer", + "task_type": "classification", + "input_splits": { + "train": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "val": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "test": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/staged_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/staged_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..66c2930e6305d84ec5097ebe6ee91ea7549f3bf6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/staged_input_manifest.json @@ -0,0 +1,174 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/public_gate/public_gate_report.json", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..8a1a806f0d0925a39f8520d080fff5c6c789d69a --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/runtime_result.json @@ -0,0 +1,15 @@ +{ + "dataset_id": "c6", + "model": "bayesnet", + "run_id": "bayesnet-c6-20260422_060152", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "success", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet-c6-7636-20260422_060304.csv", + "model_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_model.pkl" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/bayesnet/adapter_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/bayesnet/adapter_report.json new file mode 100644 index 0000000000000000000000000000000000000000..9133af59dca39df3be2a6d1018865a85d053cc67 --- /dev/null +++ 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Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" 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Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1275,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1253,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +166,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,396,0,Advanced,Integration,Integration Techniques,Power of trigonometric functions +1044,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1173,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1487,Portugal,1186,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Partial fractions decomposition" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector 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number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear 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+91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +473,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +633,Italy,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +80,Lithuania,532,1,Basic,Integration,Double Integration,"Cartesian coordinates,Iterate integrals,x-simple region (type II)" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +509,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/staged/public/train.csv 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+1322,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +414,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1533,Portugal,561,0,Basic,Analytic 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+91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1363,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1299,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +988,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations 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Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" 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Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" 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Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" 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of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear 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+1028,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +913,Ireland,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1156,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +695,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +535,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1140,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +72,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1340,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1363,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +411,Portugal,160,0,Basic,Differential Equations,Differential Equations,Linear differential equation +634,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +468,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1331,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +600,Italy,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/train_20260422_060152.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/train_20260422_060152.log new file mode 100644 index 0000000000000000000000000000000000000000..678d0b5dc08208cbf0378aeb6a312718aed855a9 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/train_20260422_060152.log @@ -0,0 +1,50 @@ + +========== +== CUDA == +========== + +CUDA Version 12.8.1 + +Container image Copyright (c) 2016-2023, NVIDIA CORPORATION & AFFILIATES. All rights reserved. + +This container image and its contents are governed by the NVIDIA Deep Learning Container License. +By pulling and using the container, you accept the terms and conditions of this license: +https://developer.nvidia.com/ngc/nvidia-deep-learning-container-license + +A copy of this license is made available in this container at /NGC-DL-CONTAINER-LICENSE for your convenience. + +WARNING: The NVIDIA Driver was not detected. GPU functionality will not be available. + Use the NVIDIA Container Toolkit to start this container with GPU support; see + https://docs.nvidia.com/datacenter/cloud-native/ . + +/usr/local/lib/python3.10/dist-packages/pgmpy/estimators/__init__.py:4: FutureWarning: `pgmpy.estimators.StructureScore` is deprecated and will be removed in a future release. Use `pgmpy.structure_score` instead. + from .StructureScore import ( +ERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts. +synthcity 0.2.12 requires arfpy, which is not installed. +synthcity 0.2.12 requires be-great>=0.0.5; python_version >= "3.9", which is not installed. +synthcity 0.2.12 requires decaf-synthetic-data>=0.1.6, which is not installed. +synthcity 0.2.12 requires fastai<2.8, which is not installed. +synthcity 0.2.12 requires fastcore<1.8, which is not installed. +synthcity 0.2.12 requires fflows, which is not installed. +synthcity 0.2.12 requires geomloss, which is not installed. +synthcity 0.2.12 requires importlib-metadata, which is not installed. +synthcity 0.2.12 requires lifelines<0.30.0,>=0.29.0, which is not installed. +synthcity 0.2.12 requires monai, which is not installed. +synthcity 0.2.12 requires nflows>=0.14, which is not installed. +synthcity 0.2.12 requires opacus>=1.3, which is not installed. +synthcity 0.2.12 requires pycox, which is not installed. +synthcity 0.2.12 requires pykeops, which is not installed. +synthcity 0.2.12 requires redis, which is not installed. +synthcity 0.2.12 requires shap, which is not installed. +synthcity 0.2.12 requires tenacity, which is not installed. +synthcity 0.2.12 requires tsai; python_version > "3.7", which is not installed. +synthcity 0.2.12 requires xgbse>=0.3.1, which is not installed. +synthcity 0.2.12 requires networkx<3.0,>2.0, but you have networkx 3.4.2 which is incompatible. +synthcity 0.2.12 requires numpy<2.0,>=1.20, but you have numpy 2.2.6 which is incompatible. +synthcity 0.2.12 requires pgmpy<1.0, but you have pgmpy 1.1.0 which is incompatible. +synthcity 0.2.12 requires torch<2.3,>=2.1, but you have torch 2.8.0+cu128 which is incompatible. +synthcity 0.2.12 requires xgboost<3.0.0, but you have xgboost 3.2.0 which is incompatible. +WARNING: Running pip as the 'root' user can result in broken permissions and conflicting behaviour with the system package manager, possibly rendering your system unusable. It is recommended to use a virtual environment instead: https://pip.pypa.io/warnings/venv. Use the --root-user-action option if you know what you are doing and want to suppress this warning. +[BayesNet] max_bins=10 (cols_in_df=8, rows=7636) +[BayesNet] Training on 7636 rows, 8 cols (encoded) +[BayesNet] Model saved -> /work/output-SpecializedModels/c6/bayesnet/bayesnet-c6-20260422_060152/bayesnet_model.pkl diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/_ctgan_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/_ctgan_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..079de08754d4cefd9661c4b86244f5000fdf4ad6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/_ctgan_generate.py @@ -0,0 +1,18 @@ +import sys +sys.path.insert(0, "/work") +from src.SpecificModels.ctgan_rdt_inverse_fix import apply_ctgan_inverse_fix +apply_ctgan_inverse_fix() +import pandas as pd +from ctgan.synthesizers.ctgan import CTGAN +model = CTGAN.load("/work/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/models_300epochs/ctgan_300epochs.pt") +total = 7636 +chunk = min(50000, total) if total > 50000 else total +parts = [] +left = total +while left > 0: + take = min(chunk, left) + parts.append(model.sample(take)) + left -= take +sampled = pd.concat(parts, ignore_index=True) if len(parts) > 1 else parts[0] +sampled.to_csv("/work/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan-c6-7636-20260422_030517.csv", index=False) +print("[CTGAN] Generated", total, "rows in", len(parts), "chunks ->", "/work/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan-c6-7636-20260422_030517.csv") \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan_metadata.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan_metadata.json new file mode 100644 index 0000000000000000000000000000000000000000..fd44f1730c28205b8349695531f19bedb2470087 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan_metadata.json @@ -0,0 +1,36 @@ +{ + "columns": [ + { + "name": "Student ID", + "type": "continuous" + }, + { + "name": "Student Country", + "type": "categorical" + }, + { + "name": "Question ID", + "type": "continuous" + }, + { + "name": "Type of Answer", + "type": "categorical" + }, + { + "name": "Question Level", + "type": "categorical" + }, + { + "name": "Topic", + "type": "categorical" + }, + { + "name": "Subtopic", + "type": "categorical" + }, + { + "name": "Keywords", + "type": "categorical" + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/gen_20260422_030517.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/gen_20260422_030517.log new file mode 100644 index 0000000000000000000000000000000000000000..b98e22464875258b657f2b299393dfd0d92e20a0 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/gen_20260422_030517.log @@ -0,0 +1,2 @@ +[CTGAN] Generated 7636 rows in 1 chunks -> /work/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan-c6-7636-20260422_030517.csv +[W421 19:05:28.003895641 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..6168765e31eaf2f2f4406b02a88b7b33fb522782 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + "model": "ctgan", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "exists": true, + "size": 849500, + "sha256": "7d8f85a52de0e63e292778c26cb06223383b366c589d4226c3de68b111ba5272" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "exists": true, + "size": 108137, + "sha256": "9ede9f1e2036e743d822e8ed8d7b5e1050159e8fc7b402b758a294f7a14528fe" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv", + "exists": true, + "size": 107696, + "sha256": "d28b60b361526450f0c203ddf50498854cb66ad5c1978516a99c265f529f8e4f" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_profile.json", + "exists": true, + "size": 4145, + "sha256": "70c4d3f4f544b9bff7543f502136d9b1403d8589ad5ef0a9695842d8ef9d5185" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_contract_v1.json", + "exists": true, + "size": 4740, + "sha256": "602750e8159221cf97836d44d530098411b5f2cd6fc47c06776171da79d06593" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/models_300epochs/train_20260422_025941.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/models_300epochs/train_20260422_025941.log new file mode 100644 index 0000000000000000000000000000000000000000..a5bb93870d124a1b6741bcb21e2b106f4481c213 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/models_300epochs/train_20260422_025941.log @@ -0,0 +1,11 @@ +/opt/conda/lib/python3.11/site-packages/torch/autograd/graph.py:841: UserWarning: Attempting to run cuBLAS, but there was no current CUDA context! Attempting to set the primary context... (Triggered internally at /pytorch/aten/src/ATen/cuda/CublasHandlePool.cpp:270.) + return Variable._execution_engine.run_backward( # Calls into the C++ engine to run the backward pass +[W421 19:05:13.975666541 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:13.992780049 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:13.005402886 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:13.007821478 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:13.020639115 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:14.049323339 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:14.080992950 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:14.134090679 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) +[W421 19:05:16.216594021 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..62de672c6242a8fe33594746bc19ea183e203ce7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,169 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "columns": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/public_gate_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68d5e4a7e5533d3434a96c463bb1367c9d25256e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "c6", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "Type of Answer", + "task_type": "classification", + "input_splits": { + "train": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "val": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "test": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/staged_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/staged_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..236d7c4c0060068f5fa4641dba42b33c06a40151 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/staged_input_manifest.json @@ -0,0 +1,174 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/public_gate_report.json", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..199e4de8aa965b09c35e1f03fb785289f6b29ff7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/runtime_result.json @@ -0,0 +1,15 @@ +{ + "dataset_id": "c6", + "model": "ctgan", + "run_id": "ctgan-c6-20260422_025941", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "success", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/ctgan-c6-7636-20260422_030517.csv", + "model_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/models_300epochs/ctgan_300epochs.pt" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/ctgan/adapter_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/ctgan/adapter_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68880db9a138b935b0fdc5a77aaf3431d0a76541 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/ctgan/adapter_report.json @@ -0,0 +1,7 @@ +{ + "adapter_ready_status": "pass", + "adapter_fail_reason_code": null, + "adapter_fail_detail": null, + "adapter_transforms_applied": [], + "model_input_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/ctgan/model_input_manifest.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/ctgan/adapter_transforms_applied.json 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"c6", + "model": "ctgan", + "target_column": "Type of Answer", + "task_type": "classification", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ], + "public_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/staged_input_manifest.json", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/staged_features.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/staged_features.json new file mode 100644 index 0000000000000000000000000000000000000000..5523fb7a16718db5c4122f8381d3cff51e337c45 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/staged_features.json @@ -0,0 +1,42 @@ +[ + { + "feature_name": "Student ID", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "Student Country", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Question ID", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "Type of Answer", + "data_type": "binary", + "is_target": true + }, + { + "feature_name": "Question Level", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Topic", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Subtopic", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Keywords", + "data_type": "categorical", + "is_target": false + } +] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/test.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/test.csv new file mode 100644 index 0000000000000000000000000000000000000000..0cfdbd1a841b9ae79ea9c497d5582a798326bb4f --- /dev/null +++ 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+591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,542,1,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +128,Lithuania,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,779,1,Advanced,Linear 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Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +885,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +899,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" 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Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,396,0,Advanced,Integration,Integration Techniques,Power of trigonometric functions +1044,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1173,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" 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linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +510,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +167,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +473,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +633,Italy,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +80,Lithuania,532,1,Basic,Integration,Double Integration,"Cartesian coordinates,Iterate integrals,x-simple region (type II)" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +509,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear 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linear transformation +626,Italy,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +57,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1219,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +634,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,462,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +876,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1171,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +451,Portugal,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +551,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +697,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +657,Ireland,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1420,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +851,Italy,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +840,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +633,Italy,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +988,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1331,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1363,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1299,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +988,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations 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Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" 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Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +405,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +697,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1274,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1533,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/ctgan/ctgan-c6-20260422_025941/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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application,Linearity" +1299,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +375,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1028,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +913,Ireland,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1156,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +695,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +535,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1140,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +72,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1340,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1363,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +411,Portugal,160,0,Basic,Differential Equations,Differential Equations,Linear differential equation +634,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" 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Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions 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Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace 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combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" 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Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First 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+483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..58ff5572451edd227f05d26c73d49bbccbe04282 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + "model": "realtabformer", + "inputs": { + "train_csv": { + "path": 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b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/gen_20260330_005111.log new file mode 100644 index 0000000000000000000000000000000000000000..660e17dfb901a8d25444eba6cd8701e8de758868 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/gen_20260330_005111.log @@ -0,0 +1,13 @@ + + 0%| | 0/1000 [00:00 /work/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/rtf-c6-1000-20260330_005111.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/gen_20260418_073621.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/gen_20260418_073621.log new file mode 100644 index 0000000000000000000000000000000000000000..c0635d90a068a4879b7c1657e4dcec90f1f99ab7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/gen_20260418_073621.log @@ -0,0 +1,11 @@ +Traceback (most recent call last): + File "", line 198, in _run_module_as_main + File "", line 88, in _run_code + File "/tmp/realtabformer/src/realtabformer/benchmark_cli.py", line 105, in + main() + File "/tmp/realtabformer/src/realtabformer/benchmark_cli.py", line 75, in main + load_path = _find_latest_model_dir(model_dir) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + File "/tmp/realtabformer/src/realtabformer/benchmark_cli.py", line 29, in _find_latest_model_dir + raise FileNotFoundError( +FileNotFoundError: No id* subdir in: /work/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/models_100epochs/id000017748030670275743744. 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b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68d5e4a7e5533d3434a96c463bb1367c9d25256e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "c6", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "Type of Answer", + "task_type": "classification", + "input_splits": { + 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Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +600,Italy,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1146,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +419,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1322,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +26,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +986,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +551,Lithuania,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1505,Portugal,280,1,Basic,Optimization,Linear Optimization,Linear programming +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear 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Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +885,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +899,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1219,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +457,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +855,Italy,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1275,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1253,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +166,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,396,0,Advanced,Integration,Integration Techniques,Power of trigonometric functions +1044,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1173,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1487,Portugal,1186,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Partial fractions decomposition" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +753,Italy,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1192,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +854,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +510,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +167,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +473,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +633,Italy,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +80,Lithuania,532,1,Basic,Integration,Double Integration,"Cartesian coordinates,Iterate integrals,x-simple region (type II)" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +509,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear 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Techniques,Trigonometric substitution +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1533,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +982,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1350,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +712,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +52,Portugal,463,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +966,Portugal,438,0,Basic,Linear Algebra,Vector 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Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1029,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +840,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1505,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +890,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" 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+876,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +451,Portugal,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +551,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +697,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +657,Ireland,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1420,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +851,Italy,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +840,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +633,Italy,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +988,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1331,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1363,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1299,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +988,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +633,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1334,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation 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Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +405,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +697,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1274,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1533,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/rtf-c6-20260329_231509/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +52,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +898,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +394,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1071,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +72,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +690,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +411,Portugal,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +378,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +689,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1286,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +462,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1256,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1299,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +375,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1028,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +913,Ireland,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1156,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +695,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +535,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1140,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +72,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1340,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1363,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +411,Portugal,160,0,Basic,Differential Equations,Differential Equations,Linear differential equation +634,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +468,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1331,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +600,Italy,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector 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independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,542,1,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +128,Lithuania,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear 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Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear 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application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +633,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1334,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +483,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1109,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +751,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +988,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +898,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +405,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +697,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1274,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1533,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/realtabformer/staged/public/val.csv @@ 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part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear 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Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real 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Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector 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+406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear 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"semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ], + "public_manifest": "output-SpecializedModels/c6/realtabformer/public_gate/staged_input_manifest.json", + "train_csv": "output-SpecializedModels/c6/realtabformer/staged/public/train.csv", + "val_csv": "output-SpecializedModels/c6/realtabformer/staged/public/val.csv", + "test_csv": "output-SpecializedModels/c6/realtabformer/staged/public/test.csv", + "features_json": "output-SpecializedModels/c6/realtabformer/staged/public/staged_features.json", + "public_gate_report": "output-SpecializedModels/c6/realtabformer/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/_tabpfgen_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/_tabpfgen_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..81c455fceb9b5f12ce687ca301ed5ec568530e91 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/_tabpfgen_generate.py @@ -0,0 +1,87 @@ +import numpy as np +import pandas as pd +import json +from tabpfgen import TabPFGen + +df = pd.read_csv("/work/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/train.csv") +target_col = "Type of Answer" + +feature_cols = [c for c in df.columns if c != target_col] + +# --- Label-encode categorical / object columns --- +cat_encodings = {} # col -> list of unique values (index = code) +for col in feature_cols: + if df[col].dtype == object or str(df[col].dtype) == 'category': + cats = sorted(df[col].dropna().unique().tolist(), key=str) + cat_map = {v: i for i, v in enumerate(cats)} + df[col] = df[col].map(cat_map).astype(float) + cat_encodings[col] = cats + print(f"[TabPFGen] Label-encoded '{col}' ({len(cats)} categories)") + +# Encode target if categorical +target_cats = None +if df[target_col].dtype == object or str(df[target_col].dtype) == 'category': + cats = sorted(df[target_col].dropna().unique().tolist(), key=str) + t_map = {v: i for i, v in enumerate(cats)} + df[target_col] = df[target_col].map(t_map).astype(float) + target_cats = cats + print(f"[TabPFGen] Label-encoded target '{target_col}' ({len(cats)} categories)") + +X = df[feature_cols].values.astype(np.float32) +y = df[target_col].values +target_n = int(7636) + +# Handle NaN +for i in range(X.shape[1]): + col_vals = X[:, i] + mask = np.isnan(col_vals) + if mask.any(): + mean_val = np.nanmean(col_vals) + X[mask, i] = mean_val if not np.isnan(mean_val) else 0.0 + +gen = TabPFGen( + n_sgld_steps=1000, + sgld_step_size=0.01, + sgld_noise_scale=0.01, + device="auto", +) + +print(f"[TabPFGen] Generating {target_n} rows via generate_classification") +X_syn, y_syn = gen.generate_classification(X, y, n_samples=target_n) + +syn_df = pd.DataFrame(X_syn, columns=feature_cols) +syn_df[target_col] = y_syn + +# --- Inverse label-encoding for categorical columns --- +for col, cats in cat_encodings.items(): + # Round to nearest integer index, clamp to valid range + codes = np.round(syn_df[col].values).astype(int) + codes = np.clip(codes, 0, len(cats) - 1) + syn_df[col] = [cats[c] for c in codes] + +if target_cats is not None: + codes = np.round(syn_df[target_col].values).astype(int) + codes = np.clip(codes, 0, len(target_cats) - 1) + syn_df[target_col] = [target_cats[c] for c in codes] + +# Ensure output row count is strictly aligned with target_n. +if len(syn_df) > target_n: + print(f"[TabPFGen] Trimming rows: {len(syn_df)} -> {target_n}") + syn_df = syn_df.iloc[:target_n].copy() +elif len(syn_df) < target_n: + deficit = target_n - len(syn_df) + print(f"[TabPFGen] Padding rows: {len(syn_df)} -> {target_n} (deficit={deficit})") + if len(syn_df) > 0: + extra = syn_df.sample(n=deficit, replace=True, random_state=42) + syn_df = pd.concat([syn_df.reset_index(drop=True), extra.reset_index(drop=True)], ignore_index=True) + else: + # Defensive fallback: if generator returns empty, bootstrap from training rows. + syn_df = df[feature_cols + [target_col]].sample( + n=target_n, replace=True, random_state=42 + ).reset_index(drop=True) + +syn_df = syn_df[list(df.columns)] +if len(syn_df) != target_n: + raise RuntimeError(f"[TabPFGen] Row alignment failed: got {len(syn_df)}, expected {target_n}") +syn_df.to_csv("/work/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen-c6-7636-20260422_200031.csv", index=False) +print(f"[TabPFGen] Saved {len(syn_df)} rows -> /work/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen-c6-7636-20260422_200031.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/gen_20260422_200031.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/gen_20260422_200031.log new file mode 100644 index 0000000000000000000000000000000000000000..4a90f725fcf386bc006c01497df85c52153c333b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/gen_20260422_200031.log @@ -0,0 +1,18 @@ +[TabPFGen] Label-encoded 'Student Country' (8 categories) +[TabPFGen] Label-encoded 'Question Level' (2 categories) +[TabPFGen] Label-encoded 'Topic' (14 categories) +[TabPFGen] Label-encoded 'Subtopic' (24 categories) +[TabPFGen] Label-encoded 'Keywords' (360 categories) +[TabPFGen] Generating 7636 rows via generate_classification +Step 0/1000 +Step 100/1000 +Step 200/1000 +Step 300/1000 +Step 400/1000 +Step 500/1000 +Step 600/1000 +Step 700/1000 +Step 800/1000 +Step 900/1000 +[TabPFGen] Saved 7636 rows -> /work/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen-c6-7636-20260422_200031.csv +[W422 12:01:18.057859532 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..0c7e627ff39f71f89cfcfe7d5c87dd7ae0e7a960 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + "model": "tabpfgen", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "exists": true, + "size": 849500, + "sha256": "7d8f85a52de0e63e292778c26cb06223383b366c589d4226c3de68b111ba5272" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "exists": true, + "size": 108137, + "sha256": "9ede9f1e2036e743d822e8ed8d7b5e1050159e8fc7b402b758a294f7a14528fe" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv", + "exists": true, + "size": 107696, + "sha256": "d28b60b361526450f0c203ddf50498854cb66ad5c1978516a99c265f529f8e4f" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_profile.json", + "exists": true, + "size": 4145, + "sha256": "70c4d3f4f544b9bff7543f502136d9b1403d8589ad5ef0a9695842d8ef9d5185" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_contract_v1.json", + "exists": true, + "size": 4740, + "sha256": "602750e8159221cf97836d44d530098411b5f2cd6fc47c06776171da79d06593" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..62de672c6242a8fe33594746bc19ea183e203ce7 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,169 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "columns": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/public_gate_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68d5e4a7e5533d3434a96c463bb1367c9d25256e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "c6", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "Type of Answer", + "task_type": "classification", + "input_splits": { + "train": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "val": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "test": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/staged_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/staged_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..7b25db05fad1406ffa89a01909d8f790dc89033d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/staged_input_manifest.json @@ -0,0 +1,174 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/public_gate_report.json", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..242c9893fef89843aaf62292deaf39590371a33c --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/runtime_result.json @@ -0,0 +1,15 @@ +{ + "dataset_id": "c6", + "model": "tabpfgen", + "run_id": "tabpfgen-c6-20260422_200030", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "success", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen-c6-7636-20260422_200031.csv", + "model_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/staged_features.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/staged_features.json new file mode 100644 index 0000000000000000000000000000000000000000..5523fb7a16718db5c4122f8381d3cff51e337c45 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/staged_features.json @@ -0,0 +1,42 @@ +[ + { + "feature_name": "Student ID", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "Student Country", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Question ID", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "Type of Answer", + "data_type": "binary", + "is_target": true + }, + { + "feature_name": "Question Level", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Topic", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Subtopic", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Keywords", + "data_type": "categorical", + "is_target": false + } +] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/test.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/test.csv new file mode 100644 index 0000000000000000000000000000000000000000..0cfdbd1a841b9ae79ea9c497d5582a798326bb4f --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/test.csv @@ -0,0 +1,957 @@ +Student ID,Student Country,Question ID,Type of Answer,Question Level,Topic,Subtopic,Keywords +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" 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Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +203,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +641,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +958,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector 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+1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +340,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +176,Italy,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1252,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1157,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,485,1,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +351,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +419,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +71,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +81,Lithuania,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +693,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +599,Italy,601,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1517,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +524,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +376,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +664,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,503,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +174,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +675,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +524,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +592,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +394,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +600,Italy,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1146,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +419,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1322,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +26,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +986,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +551,Lithuania,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1505,Portugal,280,1,Basic,Optimization,Linear Optimization,Linear programming +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,542,1,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +128,Lithuania,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +552,Portugal,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +979,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +982,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1420,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +378,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +885,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +899,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1219,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +457,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +855,Italy,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1275,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1253,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +166,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,396,0,Advanced,Integration,Integration Techniques,Power of trigonometric functions +1044,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1173,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1487,Portugal,1186,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Partial fractions decomposition" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +753,Italy,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1192,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +854,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +510,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +167,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution 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Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/train.csv 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type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +633,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1334,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +483,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1109,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +751,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +988,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +898,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +405,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +697,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1274,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1533,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" 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+552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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+1028,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +913,Ireland,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1156,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +695,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +535,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1140,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +72,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1340,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1363,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +411,Portugal,160,0,Basic,Differential Equations,Differential Equations,Linear differential equation +634,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" 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Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/adapter_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/adapter_report.json new file mode 100644 index 0000000000000000000000000000000000000000..d235ea9404787f536c0b249809f9182c3e8af065 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/adapter_report.json @@ -0,0 +1,7 @@ +{ + "adapter_ready_status": "pass", + "adapter_fail_reason_code": null, + "adapter_fail_detail": null, + "adapter_transforms_applied": [], + "model_input_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/model_input_manifest.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/adapter_transforms_applied.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/adapter_transforms_applied.json new file mode 100644 index 0000000000000000000000000000000000000000..0637a088a01e8ddab3bf3fa98dbe804cbde1a0dc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/adapter_transforms_applied.json @@ -0,0 +1 @@ +[] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/model_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..9d1e41dc7fa8118b59dec275dde6e9950156a412 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/tabpfgen/model_input_manifest.json @@ -0,0 +1,176 @@ +{ + "dataset_id": "c6", + "model": "tabpfgen", + "target_column": "Type of Answer", + "task_type": "classification", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ], + "public_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/staged_input_manifest.json", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen_meta.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen_meta.json new file mode 100644 index 0000000000000000000000000000000000000000..2447cf3a09e60e72f96bbd1be477198ae1d3847d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen_meta.json @@ -0,0 +1,8 @@ +{ + "csv_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/train.csv", + "json_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/staged/public/staged_features.json", + "target_col": "Type of Answer", + "is_classification": true, + "n_rows": 7636, + "n_cols": 8 +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/train_20260422_200031.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/train_20260422_200031.log new file mode 100644 index 0000000000000000000000000000000000000000..eeedb8f76a0c360d0e38d533cd92723fa4bc6797 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/train_20260422_200031.log @@ -0,0 +1 @@ +[TabPFGen] No training needed (pretrained). Meta saved to /data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabpfgen/tabpfgen-c6-20260422_200030/tabpfgen_meta.json diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/_tabsyn_sample.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/_tabsyn_sample.py new file mode 100644 index 0000000000000000000000000000000000000000..f51ac4576a96d5e7d155e93e3d98d7635f2ac33f --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/_tabsyn_sample.py @@ -0,0 +1,60 @@ +import os, sys, subprocess + +work_dir = "/work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426" +dataname = "benchmark_ds" +output_csv = "/work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/tabsyn-c6-7636-20260329_102858.csv" +tabsyn_root = "/workspace/tabsyn" + +assert os.path.exists(tabsyn_root), f"TabSyn source not mounted: {tabsyn_root}" + +pip_libs = "/pip_libs" +ppath = tabsyn_root + os.pathsep + pip_libs +old = os.environ.get("PYTHONPATH", "") +if old: + ppath += os.pathsep + old +os.environ["PYTHONPATH"] = ppath +sys.path.insert(0, pip_libs) +sys.path.insert(0, tabsyn_root) + +try: + import icecream +except ModuleNotFoundError: + print("[TabSyn] Installing dependencies (first run, cached for later)...") + subprocess.run( + [sys.executable, "-m", "pip", "install", "--target", pip_libs, + "icecream", "category_encoders", "tomli", "tomli-w", + "imbalanced-learn", "ml_collections", "rdt", + "xgboost", "xlrd", "openpyxl", "numpy<2", "-q"], + check=True + ) + subprocess.run( + [sys.executable, "-m", "pip", "install", "--target", pip_libs, + "--no-deps", "libzero", "-q"], + check=True + ) + +os.chdir(tabsyn_root) + +# Ensure data symlink exists +data_link = os.path.join(tabsyn_root, "data", dataname) +data_src = os.path.join(work_dir, "data", dataname) +os.makedirs(os.path.join(tabsyn_root, "data"), exist_ok=True) +if os.path.exists(data_link): + os.remove(data_link) +os.symlink(data_src, data_link) + +print(f"[TabSyn] Sampling 7636 rows") +env = os.environ.copy() +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "sample", + "--method", "tabsyn", + "--gpu", "0", + "--save_path", output_csv], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + sys.exit(ret.returncode) +print(f"[TabSyn] Saved -> {output_csv}") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/_tabsyn_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/_tabsyn_train.py new file mode 100644 index 0000000000000000000000000000000000000000..27dda6dd2404acd9ef430771066d892203e28d90 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/_tabsyn_train.py @@ -0,0 +1,83 @@ +import os, sys, subprocess + +work_dir = "/work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426" +dataname = "benchmark_ds" +tabsyn_root = "/workspace/tabsyn" + +assert os.path.exists(tabsyn_root), f"TabSyn source not mounted: {tabsyn_root}" + +pip_libs = "/pip_libs" +# Add pip_libs to PYTHONPATH so cached packages are found +ppath = tabsyn_root + os.pathsep + pip_libs +old = os.environ.get("PYTHONPATH", "") +if old: + ppath += os.pathsep + old +os.environ["PYTHONPATH"] = ppath +sys.path.insert(0, pip_libs) +sys.path.insert(0, tabsyn_root) + +# Install missing deps into persistent cache (first run only) +try: + import icecream +except ModuleNotFoundError: + print("[TabSyn] Installing dependencies (first run, cached for later)...") + # Step 1: main packages (with numpy<2 to avoid torch conflict) + subprocess.run( + [sys.executable, "-m", "pip", "install", "--target", pip_libs, + "icecream", "category_encoders", "tomli", "tomli-w", + "imbalanced-learn", "ml_collections", "rdt", + "xgboost", "xlrd", "openpyxl", "numpy<2", "-q"], + check=True + ) + # Step 2: libzero with --no-deps (avoids dependency conflict) + subprocess.run( + [sys.executable, "-m", "pip", "install", "--target", pip_libs, + "--no-deps", "libzero", "-q"], + check=True + ) + +os.chdir(tabsyn_root) + +# Symlink data dir into TabSyn data/ +data_link = os.path.join(tabsyn_root, "data", dataname) +data_src = os.path.join(work_dir, "data", dataname) +os.makedirs(os.path.join(tabsyn_root, "data"), exist_ok=True) +if os.path.exists(data_link): + os.remove(data_link) +os.symlink(data_src, data_link) + +env = os.environ.copy() + +# Data preprocessing is done on the host side (_prepare_data_dir) +# which creates .npy files, train/test CSVs, and info.json + +# Step 1: Train VAE (produces latent embeddings) +print(f"[TabSyn] Step 1/2: Training VAE in {tabsyn_root}, dataname={dataname}") +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "train", + "--method", "vae", + "--gpu", "0"], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + print("[TabSyn] VAE training failed") + sys.exit(ret.returncode) + +# Step 2: Train diffusion model on latent space +print(f"[TabSyn] Step 2/2: Training diffusion model") +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "train", + "--method", "tabsyn", + "--gpu", "0"], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + print("[TabSyn] Diffusion training failed") + sys.exit(ret.returncode) +print("[TabSyn] Training complete (VAE + Diffusion)") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_cat_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_cat_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..5027c2aead799748583a614c6cac6d2344c9ba83 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_cat_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_num_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_num_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..1c99f05b1f3fa96bf80641321d731579d41742c8 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_num_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_num_train.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_num_train.npy new file mode 100644 index 0000000000000000000000000000000000000000..43c60b4c0b4fb06068ae50c5c64e0011ff69c7c5 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/X_num_train.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/info.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/info.json new file mode 100644 index 0000000000000000000000000000000000000000..0e5417d6daea87bcdd35f36820bada9b5b8f5939 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/info.json @@ -0,0 +1,98 @@ +{ + "name": "benchmark_ds", + "task_type": "multiclass", + "n_num_features": 2, + "n_cat_features": 5, + "train_size": 8590, + "num_col_idx": [ + 0, + 2 + ], + "cat_col_idx": [ + 1, + 4, + 5, + 6, + 7 + ], + "target_col_idx": [ + 3 + ], + "column_names": [ + "Student ID", + "Student Country", + "Question ID", + "Type of Answer", + "Question Level", + "Topic", + "Subtopic", + "Keywords" + ], + "train_num": 8590, + "test_num": 956, + "header": 0, + "file_type": "csv", + "data_path": "data/benchmark_ds/train.csv", + "test_path": null, + "idx_mapping": { + "0": 0, + "1": 2, + "2": 1, + "3": 7, + "4": 3, + "5": 4, + "6": 5, + "7": 6 + }, + "inverse_idx_mapping": { + "0": 0, + "2": 1, + "1": 2, + "7": 3, + "3": 4, + "4": 5, + "5": 6, + "6": 7 + }, + "idx_name_mapping": { + "0": "Student ID", + "1": "Student Country", + "2": "Question ID", + "3": "Type of Answer", + "4": "Question Level", + "5": "Topic", + "6": "Subtopic", + "7": "Keywords" + }, + "n_classes": 2, + "metadata": { + "columns": { + "0": { + "sdtype": "numerical", + "computer_representation": "Float" + }, + "2": { + "sdtype": "numerical", + "computer_representation": "Float" + }, + "1": { + "sdtype": "categorical" + }, + "4": { + "sdtype": "categorical" + }, + "5": { + "sdtype": "categorical" + }, + "6": { + "sdtype": "categorical" + }, + "7": { + "sdtype": "categorical" + }, + "3": { + "sdtype": "categorical" + } + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/test.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/test.csv new file mode 100644 index 0000000000000000000000000000000000000000..a063e51e4e3e2d345c22f751a79d6fcca5d667b1 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/data/benchmark_ds/test.csv @@ -0,0 +1,957 @@ +Student ID,Student Country,Question ID,Type of Answer,Question Level,Topic,Subtopic,Keywords +1023,2,429,1,1,2,2,10 +692,0,204,1,1,8,10,36 +955,2,435,0,1,2,2,104 +57,0,184,1,1,8,10,85 +598,1,134,1,1,2,13,77 +986,0,430,1,0,2,2,10 +415,0,336,0,1,0,0,136 +37,0,608,0,1,2,19,163 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/dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/gen_20260329_102858.log @@ -0,0 +1,37 @@ + +============= +== PyTorch == +============= + +NVIDIA Release 25.01 (build 134983853) +PyTorch Version 2.6.0a0+ecf3bae +Container image Copyright (c) 2025, NVIDIA CORPORATION & AFFILIATES. All rights reserved. +Copyright (c) 2014-2024 Facebook Inc. +Copyright (c) 2011-2014 Idiap Research Institute (Ronan Collobert) +Copyright (c) 2012-2014 Deepmind Technologies (Koray Kavukcuoglu) +Copyright (c) 2011-2012 NEC Laboratories America (Koray Kavukcuoglu) +Copyright (c) 2011-2013 NYU (Clement Farabet) +Copyright (c) 2006-2010 NEC Laboratories America (Ronan Collobert, Leon Bottou, Iain Melvin, Jason Weston) +Copyright (c) 2006 Idiap Research Institute (Samy Bengio) +Copyright (c) 2001-2004 Idiap Research Institute (Ronan Collobert, Samy Bengio, Johnny Mariethoz) +Copyright (c) 2015 Google Inc. +Copyright (c) 2015 Yangqing Jia +Copyright (c) 2013-2016 The Caffe contributors +All rights reserved. + +Various files include modifications (c) NVIDIA CORPORATION & AFFILIATES. All rights reserved. + +This container image and its contents are governed by the NVIDIA Deep Learning Container License. +By pulling and using the container, you accept the terms and conditions of this license: +https://developer.nvidia.com/ngc/nvidia-deep-learning-container-license + +NOTE: The SHMEM allocation limit is set to the default of 64MB. This may be + insufficient for PyTorch. NVIDIA recommends the use of the following flags: + docker run --gpus all --ipc=host --ulimit memlock=-1 --ulimit stack=67108864 ... + +[TabSyn] Sampling 7636 rows +No NaNs in numerical features, skipping +(8590, 6) +Time: 1.6342849731445312 +Saving sampled data to /work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/tabsyn-c6-7636-20260329_102858.csv +[TabSyn] Saved -> /work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/tabsyn-c6-7636-20260329_102858.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..2487ed51fc0775513db9a5f1a5cc2489dbb17e93 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", 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"Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/public_gate/public_gate_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..68d5e4a7e5533d3434a96c463bb1367c9d25256e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "c6", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "Type of Answer", + "task_type": "classification", + "input_splits": { + "train": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "val": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "test": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/public_gate/staged_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/public_gate/staged_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..ad5c600d86b0808a6b8c6c457675e48722f8a76f --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/public_gate/staged_input_manifest.json @@ -0,0 +1,174 @@ +{ + "dataset_id": "c6", + "target_column": "Type of Answer", + "task_type": "classification", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/staged_features.json", + "public_gate_report": 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+ "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": 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{ + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..55136bffa18bc9aaccae54d88bba305415e0db6e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/runtime_result.json @@ -0,0 +1,15 @@ +{ + "dataset_id": "c6", + "model": "tabsyn", + "run_id": "tabsyn-c6-20260329_093426", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "success", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/tabsyn-c6-7636-20260329_102858.csv", + "model_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/staged_features.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/staged_features.json new file mode 100644 index 0000000000000000000000000000000000000000..5523fb7a16718db5c4122f8381d3cff51e337c45 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/staged_features.json @@ -0,0 +1,42 @@ +[ + { + "feature_name": "Student ID", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "Student Country", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Question ID", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "Type of Answer", + "data_type": "binary", + "is_target": true + }, + { + "feature_name": "Question Level", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Topic", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Subtopic", + "data_type": "categorical", + "is_target": false + }, + { + "feature_name": "Keywords", + "data_type": "categorical", + "is_target": false + } +] \ No newline at end of file diff --git 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Spaces,"Basis,Dimension" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1525,Portugal,770,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +91,Lithuania,509,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +340,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +176,Italy,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1252,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1157,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,485,1,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +351,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +419,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +71,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +81,Lithuania,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +693,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +599,Italy,601,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1517,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +524,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +376,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +664,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,503,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +174,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +675,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +524,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +592,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +394,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +600,Italy,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1146,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +419,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1322,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +26,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +986,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +551,Lithuania,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1505,Portugal,280,1,Basic,Optimization,Linear Optimization,Linear programming +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" 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Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear 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+525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,542,1,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +128,Lithuania,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +552,Portugal,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +979,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +982,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1420,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +378,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +885,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +899,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1219,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +457,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +855,Italy,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1275,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1253,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +166,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,396,0,Advanced,Integration,Integration Techniques,Power of trigonometric functions +1044,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1173,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1487,Portugal,1186,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Partial fractions decomposition" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +753,Italy,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1192,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +854,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +510,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +167,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +473,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +633,Italy,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +80,Lithuania,532,1,Basic,Integration,Double Integration,"Cartesian coordinates,Iterate integrals,x-simple region (type II)" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +509,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/train.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/train.csv new file mode 100644 index 0000000000000000000000000000000000000000..7d1dcc30a8f3247be8e12d774dd1436a4ed15411 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/train.csv @@ -0,0 +1,7637 @@ +Student ID,Student Country,Question ID,Type of Answer,Question 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+712,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +52,Portugal,463,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +966,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +552,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +956,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar 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Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1505,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +890,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1011,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1299,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +36,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +979,Portugal,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +717,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +64,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +955,Lithuania,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +840,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +626,Italy,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +57,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1219,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +634,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,462,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +876,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1171,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +451,Portugal,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +551,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +697,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +657,Ireland,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1420,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +851,Italy,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +840,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +633,Italy,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +988,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1331,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" 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Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" 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form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +633,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1334,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +483,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1109,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +751,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +988,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +898,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +405,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +697,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1274,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1533,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260329_093426/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +468,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1331,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +600,Italy,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" 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All rights reserved. +Copyright (c) 2014-2024 Facebook Inc. +Copyright (c) 2011-2014 Idiap Research Institute (Ronan Collobert) +Copyright (c) 2012-2014 Deepmind Technologies (Koray Kavukcuoglu) +Copyright (c) 2011-2012 NEC Laboratories America (Koray Kavukcuoglu) +Copyright (c) 2011-2013 NYU (Clement Farabet) +Copyright (c) 2006-2010 NEC Laboratories America (Ronan Collobert, Leon Bottou, Iain Melvin, Jason Weston) +Copyright (c) 2006 Idiap Research Institute (Samy Bengio) +Copyright (c) 2001-2004 Idiap Research Institute (Ronan Collobert, Samy Bengio, Johnny Mariethoz) +Copyright (c) 2015 Google Inc. +Copyright (c) 2015 Yangqing Jia +Copyright (c) 2013-2016 The Caffe contributors +All rights reserved. + +Various files include modifications (c) NVIDIA CORPORATION & AFFILIATES. All rights reserved. + +This container image and its contents are governed by the NVIDIA Deep Learning Container License. +By pulling and using the container, you accept the terms and conditions of this license: +https://developer.nvidia.com/ngc/nvidia-deep-learning-container-license + +NOTE: The SHMEM allocation limit is set to the default of 64MB. This may be + insufficient for PyTorch. NVIDIA recommends the use of the following flags: + docker run --gpus all --ipc=host --ulimit memlock=-1 --ulimit stack=67108864 ... + +[TabSyn] Step 1/2: Training VAE in /workspace/tabsyn, dataname=benchmark_ds +No NaNs in numerical features, skipping +self.category_embeddings.weight.shape=torch.Size([412, 4]) +self.category_embeddings.weight.shape=torch.Size([412, 4]) + + 0%| | 0/3 [00:00 {output_csv}") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/_tabsyn_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/_tabsyn_train.py new file mode 100644 index 0000000000000000000000000000000000000000..d6d78813d980b86d36f1f75c7bcf64ac3d13183d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/_tabsyn_train.py @@ -0,0 +1,62 @@ +import os, sys, subprocess + +work_dir = "/work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109" +dataname = "tabsyn_c6" +tabsyn_root = "/workspace/tabsyn" + +assert os.path.exists(tabsyn_root), f"TabSyn source not mounted: {tabsyn_root}" + +old = os.environ.get("PYTHONPATH", "") +os.environ["PYTHONPATH"] = tabsyn_root + (os.pathsep + old if old else "") +sys.path.insert(0, tabsyn_root) + +os.chdir(tabsyn_root) + +# Symlink data dir into TabSyn data/ +data_link = os.path.join(tabsyn_root, "data", dataname) +data_src = os.path.join(work_dir, "data", dataname) +os.makedirs(os.path.join(tabsyn_root, "data"), exist_ok=True) +if os.path.exists(data_link): + os.remove(data_link) +os.symlink(data_src, data_link) + +env = os.environ.copy() +env.setdefault("TABSYN_RESUME", "1") +_te = None +if _te is not None: + env["TABSYN_VAE_EPOCHS"] = str(_te) + env["TABSYN_DIFFUSION_MAX_EPOCHS"] = str(max(_te + 1, 2)) + +# Data preprocessing is done on the host side (_prepare_data_dir) +# which creates .npy files, train/test CSVs, and info.json + +# Step 1: Train VAE (produces latent embeddings) +print(f"[TabSyn] Step 1/2: Training VAE in {tabsyn_root}, dataname={dataname}") +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "train", + "--method", "vae", + "--gpu", "0"], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + print("[TabSyn] VAE training failed") + sys.exit(ret.returncode) + +# Step 2: Train diffusion model on latent space +print(f"[TabSyn] Step 2/2: Training diffusion model") +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "train", + "--method", "tabsyn", + "--gpu", "0"], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + print("[TabSyn] Diffusion training failed") + sys.exit(ret.returncode) +print("[TabSyn] Training complete (VAE + Diffusion)") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_cat_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_cat_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..5027c2aead799748583a614c6cac6d2344c9ba83 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_cat_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_num_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_num_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..1c99f05b1f3fa96bf80641321d731579d41742c8 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_num_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_num_train.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_num_train.npy new file mode 100644 index 0000000000000000000000000000000000000000..43c60b4c0b4fb06068ae50c5c64e0011ff69c7c5 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/X_num_train.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/info.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/info.json new file mode 100644 index 0000000000000000000000000000000000000000..88e65f0559027f57510c6362976c28a9a7e53597 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/info.json @@ -0,0 +1,98 @@ +{ + "name": "tabsyn_c6", + "task_type": "multiclass", + "n_num_features": 2, + "n_cat_features": 5, + "train_size": 8590, + "num_col_idx": [ + 0, + 2 + ], + "cat_col_idx": [ + 1, + 4, + 5, + 6, + 7 + ], + "target_col_idx": [ + 3 + ], + "column_names": [ + "Student ID", + "Student Country", + "Question ID", + "Type of Answer", + "Question Level", + "Topic", + "Subtopic", + "Keywords" + ], + "train_num": 8590, + "test_num": 956, + "header": 0, + "file_type": "csv", + "data_path": "data/tabsyn_c6/train.csv", + "test_path": null, + "idx_mapping": { + "0": 0, + "1": 2, + "2": 1, + "3": 7, + "4": 3, + "5": 4, + "6": 5, + "7": 6 + }, + "inverse_idx_mapping": { + "0": 0, + "2": 1, + "1": 2, + "7": 3, + "3": 4, + "4": 5, + "5": 6, + "6": 7 + }, + "idx_name_mapping": { + "0": "Student ID", + "1": "Student Country", + "2": "Question ID", + "3": "Type of Answer", + "4": "Question Level", + "5": "Topic", + "6": "Subtopic", + "7": "Keywords" + }, + "n_classes": 2, + "metadata": { + "columns": { + "0": { + "sdtype": "numerical", + "computer_representation": "Float" + }, + "2": { + "sdtype": "numerical", + "computer_representation": "Float" + }, + "1": { + "sdtype": "categorical" + }, + "4": { + "sdtype": "categorical" + }, + "5": { + "sdtype": "categorical" + }, + "6": { + "sdtype": "categorical" + }, + "7": { + "sdtype": "categorical" + }, + "3": { + "sdtype": "categorical" + } + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/test.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/test.csv new file mode 100644 index 0000000000000000000000000000000000000000..a063e51e4e3e2d345c22f751a79d6fcca5d667b1 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/data/tabsyn_c6/test.csv @@ -0,0 +1,957 @@ +Student ID,Student Country,Question ID,Type of 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a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/gen_20260421_004239.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/gen_20260421_004239.log new file mode 100644 index 0000000000000000000000000000000000000000..17f8921cc37e69ea41aaff28628ca32a6ef85c6b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/gen_20260421_004239.log @@ -0,0 +1,8 @@ +[TabSyn] Sampling 7636 rows +/opt/conda/lib/python3.11/site-packages/torch/cuda/__init__.py:63: FutureWarning: The pynvml package is deprecated. Please install nvidia-ml-py instead. If you did not install pynvml directly, please report this to the maintainers of the package that installed pynvml for you. + import pynvml # type: ignore[import] +No NaNs in numerical features, skipping +(8590, 6) +Time: 3.2491841316223145 +Saving sampled data to /work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/tabsyn-c6-7636-20260421_004239.csv +[TabSyn] Saved -> /work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/tabsyn-c6-7636-20260421_004239.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..2487ed51fc0775513db9a5f1a5cc2489dbb17e93 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + 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multiplication, and total probability rules,Sample space" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +81,Lithuania,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +693,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +599,Italy,601,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1517,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +524,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +376,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +664,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,503,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +174,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +675,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +524,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +592,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +394,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +600,Italy,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1146,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +419,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1322,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +26,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +986,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +551,Lithuania,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1505,Portugal,280,1,Basic,Optimization,Linear Optimization,Linear programming +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" 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transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations 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Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +552,Portugal,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian 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+1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1219,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre 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+661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1487,Portugal,1186,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Partial fractions decomposition" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +753,Italy,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1192,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +854,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +510,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +167,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +473,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +633,Italy,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +80,Lithuania,532,1,Basic,Integration,Double Integration,"Cartesian coordinates,Iterate integrals,x-simple region (type II)" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +509,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/staged/public/train.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/staged/public/train.csv new file mode 100644 index 0000000000000000000000000000000000000000..7d1dcc30a8f3247be8e12d774dd1436a4ed15411 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/staged/public/train.csv @@ -0,0 +1,7637 @@ +Student ID,Student Country,Question ID,Type of Answer,Question 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Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1505,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +890,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1011,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1299,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +36,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +979,Portugal,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +717,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +64,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +955,Lithuania,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +840,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +626,Italy,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +57,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,922,0,Basic,Real Functions of a 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linear transformation +1080,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1171,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +451,Portugal,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +551,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +697,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +657,Ireland,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1420,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +851,Italy,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +840,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +633,Italy,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +988,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1331,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" 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Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" 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Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" 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Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" 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+203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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expressions, Equations, and Inequalities",Simplify expressions +52,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +898,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +394,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1071,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +72,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +690,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +411,Portugal,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +378,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +689,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1286,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +462,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1256,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1299,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +375,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1028,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +913,Ireland,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1156,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +695,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +535,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1140,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +72,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1340,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1363,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +411,Portugal,160,0,Basic,Differential Equations,Differential Equations,Linear differential equation +634,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of 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+64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" 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Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear 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+524,0,136,1,1,2,13,135 +633,1,140,1,0,2,13,293 +80,2,532,1,1,7,23,0 +890,0,277,0,1,2,4,25 +509,0,950,1,1,2,8,15 +955,2,290,1,0,2,4,25 +966,0,321,0,1,2,4,25 +1080,0,383,1,1,2,4,7 +1220,3,416,1,1,2,2,14 +986,0,424,1,0,2,2,2 +80,2,125,1,1,2,13,223 +1135,0,293,0,0,2,4,4 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/train_20260420_233109.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/train_20260420_233109.log new file mode 100644 index 0000000000000000000000000000000000000000..af8903f70a54aa735f5fef95aafc2355cfbfdb24 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233109/train_20260420_233109.log @@ -0,0 +1,38252 @@ +[TabSyn] Step 1/2: Training VAE in /workspace/tabsyn, dataname=tabsyn_c6 +/opt/conda/lib/python3.11/site-packages/torch/cuda/__init__.py:63: FutureWarning: The pynvml package is deprecated. Please install nvidia-ml-py instead. If you did not install pynvml directly, please report this to the maintainers of the package that installed pynvml for you. + import pynvml # type: ignore[import] +No NaNs in numerical features, skipping +self.category_embeddings.weight.shape=torch.Size([412, 4]) +self.category_embeddings.weight.shape=torch.Size([412, 4]) + + 0%| | 0/3 [00:00 {output_csv}") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/_tabsyn_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/_tabsyn_train.py new file mode 100644 index 0000000000000000000000000000000000000000..840e9d935cf98bf91297aa2cd27dcc1f16c15f6f --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/_tabsyn_train.py @@ -0,0 +1,62 @@ +import os, sys, subprocess + +work_dir = "/work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446" +dataname = "tabsyn_c6" +tabsyn_root = "/workspace/tabsyn" + +assert os.path.exists(tabsyn_root), f"TabSyn source not mounted: {tabsyn_root}" + +old = os.environ.get("PYTHONPATH", "") +os.environ["PYTHONPATH"] = tabsyn_root + (os.pathsep + old if old else "") +sys.path.insert(0, tabsyn_root) + +os.chdir(tabsyn_root) + +# Symlink data dir into TabSyn data/ +data_link = os.path.join(tabsyn_root, "data", dataname) +data_src = os.path.join(work_dir, "data", dataname) +os.makedirs(os.path.join(tabsyn_root, "data"), exist_ok=True) +if os.path.exists(data_link): + os.remove(data_link) +os.symlink(data_src, data_link) + +env = os.environ.copy() +env.setdefault("TABSYN_RESUME", "1") +_te = None +if _te is not None: + env["TABSYN_VAE_EPOCHS"] = str(_te) + env["TABSYN_DIFFUSION_MAX_EPOCHS"] = str(max(_te + 1, 2)) + +# Data preprocessing is done on the host side (_prepare_data_dir) +# which creates .npy files, train/test CSVs, and info.json + +# Step 1: Train VAE (produces latent embeddings) +print(f"[TabSyn] Step 1/2: Training VAE in {tabsyn_root}, dataname={dataname}") +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "train", + "--method", "vae", + "--gpu", "0"], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + print("[TabSyn] VAE training failed") + sys.exit(ret.returncode) + +# Step 2: Train diffusion model on latent space +print(f"[TabSyn] Step 2/2: Training diffusion model") +ret = subprocess.run( + [sys.executable, "main.py", + "--dataname", dataname, + "--mode", "train", + "--method", "tabsyn", + "--gpu", "0"], + cwd=tabsyn_root, + env=env +) +if ret.returncode != 0: + print("[TabSyn] Diffusion training failed") + sys.exit(ret.returncode) +print("[TabSyn] Training complete (VAE + Diffusion)") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_cat_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_cat_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..5027c2aead799748583a614c6cac6d2344c9ba83 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_cat_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_num_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_num_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..1c99f05b1f3fa96bf80641321d731579d41742c8 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_num_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_num_train.npy b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_num_train.npy new file mode 100644 index 0000000000000000000000000000000000000000..43c60b4c0b4fb06068ae50c5c64e0011ff69c7c5 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/X_num_train.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/info.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/info.json new file mode 100644 index 0000000000000000000000000000000000000000..88e65f0559027f57510c6362976c28a9a7e53597 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/info.json @@ -0,0 +1,98 @@ +{ + "name": "tabsyn_c6", + "task_type": "multiclass", + "n_num_features": 2, + "n_cat_features": 5, + "train_size": 8590, + "num_col_idx": [ + 0, + 2 + ], + "cat_col_idx": [ + 1, + 4, + 5, + 6, + 7 + ], + "target_col_idx": [ + 3 + ], + "column_names": [ + "Student ID", + "Student Country", + "Question ID", + "Type of Answer", + "Question Level", + "Topic", + "Subtopic", + "Keywords" + ], + "train_num": 8590, + "test_num": 956, + "header": 0, + "file_type": "csv", + "data_path": "data/tabsyn_c6/train.csv", + "test_path": null, + "idx_mapping": { + "0": 0, + "1": 2, + "2": 1, + "3": 7, + "4": 3, + "5": 4, + "6": 5, + "7": 6 + }, + "inverse_idx_mapping": { + "0": 0, + "2": 1, + "1": 2, + "7": 3, + "3": 4, + "4": 5, + "5": 6, + "6": 7 + }, + "idx_name_mapping": { + "0": "Student ID", + "1": "Student Country", + "2": "Question ID", + "3": "Type of Answer", + "4": "Question Level", + "5": "Topic", + "6": "Subtopic", + "7": "Keywords" + }, + "n_classes": 2, + "metadata": { + "columns": { + "0": { + "sdtype": "numerical", + "computer_representation": "Float" + }, + "2": { + "sdtype": "numerical", + "computer_representation": "Float" + }, + "1": { + "sdtype": "categorical" + }, + "4": { + "sdtype": "categorical" + }, + "5": { + "sdtype": "categorical" + }, + "6": { + "sdtype": "categorical" + }, + "7": { + "sdtype": "categorical" + }, + "3": { + "sdtype": "categorical" + } + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/test.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/test.csv new file mode 100644 index 0000000000000000000000000000000000000000..a063e51e4e3e2d345c22f751a79d6fcca5d667b1 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/data/tabsyn_c6/test.csv @@ -0,0 +1,957 @@ +Student ID,Student Country,Question ID,Type of Answer,Question Level,Topic,Subtopic,Keywords +1023,2,429,1,1,2,2,10 +692,0,204,1,1,8,10,36 +955,2,435,0,1,2,2,104 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Please install nvidia-ml-py instead. If you did not install pynvml directly, please report this to the maintainers of the package that installed pynvml for you. + import pynvml # type: ignore[import] +No NaNs in numerical features, skipping +(8590, 6) +Time: 2.103308916091919 +Saving sampled data to /work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/tabsyn-c6-7636-20260421_005324.csv +[TabSyn] Saved -> /work/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/tabsyn-c6-7636-20260421_005324.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..2487ed51fc0775513db9a5f1a5cc2489dbb17e93 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + 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Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +419,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1322,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +26,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +986,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +551,Lithuania,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1505,Portugal,280,1,Basic,Optimization,Linear Optimization,Linear programming +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +840,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1244,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +200,Lithuania,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +36,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1163,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1076,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +28,Portugal,390,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear 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+419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" 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frequency,Mean,Standard deviation,Variance,Median" +657,Ireland,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1420,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +851,Italy,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +840,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +633,Italy,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +988,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1331,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1363,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1299,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +988,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1011,Portugal,877,0,Basic,Differentiation,Partial Differentiation,Second order +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,285,1,Basic,Optimization,Linear Optimization,Linear programming +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +67,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +177,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations 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Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" 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Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" 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+203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1286,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +462,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1256,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1299,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +375,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1028,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +913,Ireland,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1156,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +695,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +535,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1140,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +72,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1340,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1363,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +411,Portugal,160,0,Basic,Differential Equations,Differential Equations,Linear differential equation +634,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +464,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1428,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1075,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1402,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +624,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +973,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +340,Portugal,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +634,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +576,Ireland,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +37,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +511,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1037,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +168,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +876,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1179,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +982,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +421,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1219,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +72,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +511,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1028,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1353,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +468,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +71,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1179,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +167,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +189,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +851,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1076,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +128,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +37,Portugal,176,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1276,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +393,Lithuania,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,1014,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +152,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +131,Lithuania,373,1,Advanced,Graph Theory,Graph Theory,Simple graph +405,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1420,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1247,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1340,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +72,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +913,Ireland,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +913,Ireland,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +913,Ireland,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +510,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +468,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1331,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +600,Italy,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector 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+524,0,136,1,1,2,13,135 +633,1,140,1,0,2,13,293 +80,2,532,1,1,7,23,0 +890,0,277,0,1,2,4,25 +509,0,950,1,1,2,8,15 +955,2,290,1,0,2,4,25 +966,0,321,0,1,2,4,25 +1080,0,383,1,1,2,4,7 +1220,3,416,1,1,2,2,14 +986,0,424,1,0,2,2,2 +80,2,125,1,1,2,13,223 +1135,0,293,0,0,2,4,4 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/train_20260420_233446.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/train_20260420_233446.log new file mode 100644 index 0000000000000000000000000000000000000000..2d5dcd767d02d82dcf30e3bf866ffab571320e22 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tabsyn/tabsyn-c6-20260420_233446/train_20260420_233446.log @@ -0,0 +1,45104 @@ +[TabSyn] Step 1/2: Training VAE in /workspace/tabsyn, dataname=tabsyn_c6 +/opt/conda/lib/python3.11/site-packages/torch/cuda/__init__.py:63: FutureWarning: The pynvml package is deprecated. Please install nvidia-ml-py instead. If you did not install pynvml directly, please report this to the maintainers of the package that installed pynvml for you. + import pynvml # type: ignore[import] +No NaNs in numerical features, skipping +self.category_embeddings.weight.shape=torch.Size([412, 4]) +[TabSyn][VAE] Resume from /workspace/tabsyn/tabsyn/vae/ckpt/tabsyn_c6/model.pt +self.category_embeddings.weight.shape=torch.Size([412, 4]) + + 0%| | 0/3 [00:00 50000 else total +parts = [] +left = total +while left > 0: + take = min(chunk, left) + parts.append(model.sample(take)) + left -= take +samples = pd.concat(parts, ignore_index=True) if len(parts) > 1 else parts[0] +samples.to_csv("/work/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae-c6-7636-20260419_191634.csv", index=False) +print(f"[TVAE] Generated {total} rows (chunks={len(parts)}) -> /work/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae-c6-7636-20260419_191634.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/_tvae_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/_tvae_train.py new file mode 100644 index 0000000000000000000000000000000000000000..7e96ae404dc79019e34dc5aba4de7bb8096865c3 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/_tvae_train.py @@ -0,0 +1,16 @@ +import json, sys +import pandas as pd +from ctgan.data import read_csv +from ctgan.synthesizers.tvae import TVAE + +csv_path = "/work/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/train.csv" +meta_path = "/work/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae_metadata.json" +save_path = "/work/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/models_300epochs/tvae_300epochs.pt" +epochs = 300 + +data, discrete_columns = read_csv(csv_path, meta_path, header=True, discrete=None) +print(f"[TVAE] Training on {len(data)} rows, {len(data.columns)} cols, epochs={epochs}") +model = TVAE(epochs=epochs, batch_size=500) +model.fit(data, discrete_columns) +model.save(save_path) +print(f"[TVAE] Model saved -> {save_path}") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/gen_20260419_191634.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/gen_20260419_191634.log new file mode 100644 index 0000000000000000000000000000000000000000..b7e42a3d549045788c07305cc836c7531b134a61 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/gen_20260419_191634.log @@ -0,0 +1 @@ +[TVAE] Generated 7636 rows (chunks=1) -> /work/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae-c6-7636-20260419_191634.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..f727c4c80d486ebc56fbd3a8a3d6c4815c95fc68 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "c6", + "model": "tvae", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-train.csv", + "exists": true, + "size": 849500, + "sha256": "7d8f85a52de0e63e292778c26cb06223383b366c589d4226c3de68b111ba5272" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-val.csv", + "exists": true, + "size": 108137, + "sha256": "9ede9f1e2036e743d822e8ed8d7b5e1050159e8fc7b402b758a294f7a14528fe" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/c6/c6-test.csv", + "exists": true, + "size": 107696, + "sha256": "d28b60b361526450f0c203ddf50498854cb66ad5c1978516a99c265f529f8e4f" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_profile.json", + "exists": true, + "size": 4145, + "sha256": "70c4d3f4f544b9bff7543f502136d9b1403d8589ad5ef0a9695842d8ef9d5185" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/c6/c6-dataset_contract_v1.json", + "exists": true, + "size": 4740, + "sha256": "602750e8159221cf97836d44d530098411b5f2cd6fc47c06776171da79d06593" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/models_300epochs/train_20260419_191329.log b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/models_300epochs/train_20260419_191329.log new file mode 100644 index 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b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/runtime_result.json @@ -0,0 +1,15 @@ +{ + "dataset_id": "c6", + "model": "tvae", + "run_id": "tvae-c6-20260419_191329", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "success", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae-c6-7636-20260419_191634.csv", + "model_path": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/models_300epochs/tvae_300epochs.pt" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/staged_features.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/staged_features.json new 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Spaces,"Basis,Dimension" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1525,Portugal,770,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +91,Lithuania,509,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +340,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +176,Italy,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1252,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1157,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,485,1,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +351,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +419,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +71,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +81,Lithuania,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +693,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +599,Italy,601,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1517,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +524,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +376,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +664,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,503,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +174,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +675,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +524,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +592,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +394,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +600,Italy,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1146,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +419,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1322,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +26,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +986,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +551,Lithuania,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1505,Portugal,280,1,Basic,Optimization,Linear Optimization,Linear programming +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +551,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +80,Lithuania,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +359,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +374,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1331,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1022,Lithuania,281,0,Basic,Optimization,Linear Optimization,Linear programming +71,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,942,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1044,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +69,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1076,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1557,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +591,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +972,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1332,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1143,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +473,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" 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Techniques,"Direct inspection,Simple integration" +175,Italy,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +988,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +451,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +695,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1487,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +37,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +473,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1026,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +473,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1194,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +80,Lithuania,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1517,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +1481,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +988,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +80,Lithuania,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +37,Portugal,1033,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +699,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +351,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1091,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1080,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +890,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +473,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +741,Ireland,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +426,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +955,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,687,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +693,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +639,Italy,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1156,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +633,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1080,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1247,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1044,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +174,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1420,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1243,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1411,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +387,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +582,Lithuania,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1331,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1427,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,724,1,Advanced,Integration,Integration Techniques,Trigonometric substitution +1322,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +525,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +64,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +982,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +979,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +166,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +509,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1363,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1023,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +200,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +626,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +473,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +751,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1173,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1331,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1481,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1179,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +524,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +751,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +898,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +691,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +448,Portugal,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +167,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +415,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1543,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1077,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +468,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1139,1,Basic,Set Theory,Set Theory,Subset +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1319,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +776,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +719,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +854,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +918,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +955,Lithuania,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +406,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1043,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1093,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +624,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +625,Russian Federation,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1230,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +890,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1171,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +67,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +393,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1049,Lithuania,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1444,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +174,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +52,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1288,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1525,Portugal,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +69,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1109,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +552,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1286,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +750,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +189,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +165,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +168,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,522,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +988,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +351,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,930,0,Basic,Graph Theory,Graph Theory,Planar graph +378,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +85,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +455,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +748,Portugal,386,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +885,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1379,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,410,0,Advanced,Integration,Integration Techniques,Rational functions +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +41,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,802,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1253,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1219,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +37,Portugal,719,1,Basic,Integration,Integration Techniques,Integration by parts +175,Italy,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1505,Portugal,957,0,Basic,Optimization,Linear Optimization,Linear programming +470,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +984,Romania,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +374,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +624,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +415,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +26,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +731,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +177,Italy,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +884,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1158,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +827,Italy,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +645,Italy,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1179,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +890,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +407,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +163,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +776,Portugal,702,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1292,Italy,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +510,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1229,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1080,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +720,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1022,Lithuania,958,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1271,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +524,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1272,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +380,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,487,1,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1252,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +956,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +414,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +37,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1505,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,365,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +91,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +965,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1300,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +899,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +166,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1411,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +61,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +80,Lithuania,1248,1,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates" +1191,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1135,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +524,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1350,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1163,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1284,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +155,Spain,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1375,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +415,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +85,Lithuania,1136,0,Basic,Set Theory,Set Theory,Subset +535,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +376,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +166,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +473,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +551,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1331,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1538,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1339,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +61,Portugal,172,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +683,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,314,1,Basic,Probability,Probability,"Event,Sample space" +28,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +853,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1087,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +626,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +576,Ireland,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +576,Ireland,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +419,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1185,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1350,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +695,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +657,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +972,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +661,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +72,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1146,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +359,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +414,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +85,Lithuania,1140,0,Basic,Set Theory,Set Theory,Subset +1095,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +753,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +982,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +72,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1032,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1244,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,946,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +378,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +457,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +625,Russian Federation,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1291,Italy,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +510,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +625,Russian Federation,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +28,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1077,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +524,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +890,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +359,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1557,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +913,Ireland,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +626,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +41,Portugal,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +956,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +374,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +511,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1043,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +851,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +697,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1049,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1229,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +80,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +695,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +953,Lithuania,211,1,Basic,Statistics,Statistics,Linear regression +1300,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1557,Portugal,927,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +414,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1308,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1247,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +986,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +426,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +624,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +457,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +979,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1256,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +41,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +840,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,998,1,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +71,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +218,Lithuania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +470,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +175,Italy,1493,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +41,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +651,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,672,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +470,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1375,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +741,Ireland,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1352,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1179,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +36,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +855,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +80,Lithuania,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +855,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1428,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +511,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1244,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +415,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1252,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +470,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +215,Lithuania,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1334,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +178,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +753,Italy,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1076,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +600,Italy,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +509,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1402,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1132,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1044,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +176,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +411,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +85,Lithuania,1148,1,Basic,Set Theory,Set Theory,Subset +1182,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +387,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1517,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +713,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +380,Portugal,1007,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1565,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1135,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +854,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,542,1,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +128,Lithuania,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +967,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1305,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +552,Portugal,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +979,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1077,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +982,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1420,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +378,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +885,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +899,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1324,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +667,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1219,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +457,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +855,Italy,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1275,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1253,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +166,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,396,0,Advanced,Integration,Integration Techniques,Power of trigonometric functions +1044,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1173,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +661,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1487,Portugal,1186,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Partial fractions decomposition" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +753,Italy,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1192,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +854,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +986,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +641,Italy,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1256,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1332,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +167,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1353,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1288,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +751,Lithuania,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +419,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +511,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1147,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +576,Ireland,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +414,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +510,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +167,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +473,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +633,Italy,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +80,Lithuania,532,1,Basic,Integration,Double Integration,"Cartesian coordinates,Iterate integrals,x-simple region (type II)" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +509,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1080,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/train.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/train.csv new file mode 100644 index 0000000000000000000000000000000000000000..7d1dcc30a8f3247be8e12d774dd1436a4ed15411 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/train.csv @@ -0,0 +1,7637 @@ +Student ID,Student Country,Question ID,Type of Answer,Question 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transformation,Linear application" +1126,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +69,Portugal,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +955,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" 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+712,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +473,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +52,Portugal,463,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +966,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +552,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +956,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar 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Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1505,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +890,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1011,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1299,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +36,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +979,Portugal,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +717,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +64,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +955,Lithuania,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +840,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +626,Italy,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +57,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1047,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1219,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +634,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,462,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +876,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1171,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +451,Portugal,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +551,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +697,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +657,Ireland,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1420,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +851,Italy,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +576,Ireland,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +840,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +633,Italy,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +988,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1331,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +351,Portugal,84,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +598,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" 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Differentiation,"First order,Second order" +1324,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +535,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1481,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1083,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1561,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1256,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1312,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +426,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,751,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +311,Ireland,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1081,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +985,Romania,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +71,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +80,Lithuania,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +168,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +525,Portugal,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +473,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1147,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1248,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +898,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +483,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +723,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1269,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +411,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1333,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1143,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1252,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +689,Portugal,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +548,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +81,Lithuania,468,1,Basic,Graph Theory,Graph Theory,Simple graph +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +178,Italy,753,0,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +166,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +625,Russian Federation,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +853,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1542,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +853,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1230,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +675,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" 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form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +37,Portugal,723,1,Basic,Integration,Integration Techniques,Substitution +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +176,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +426,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1252,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +422,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1269,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +406,Portugal,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1194,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1564,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +455,Portugal,850,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +753,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1517,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1518,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +468,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1197,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +624,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1321,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1024,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1481,Portugal,686,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +633,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +351,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1192,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +71,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +967,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +898,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1080,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +633,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +633,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +344,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +511,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +551,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +918,Portugal,1002,0,Basic,Differentiation,Partial Differentiation,First order +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +61,Portugal,164,1,Basic,Differential Equations,Differential Equations,Separable variables equation +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +890,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +855,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +890,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1339,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +64,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +426,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +28,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +411,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +851,Italy,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,676,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +411,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +599,Italy,618,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1363,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1281,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1158,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +466,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +675,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +71,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +511,Portugal,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +955,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +394,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +200,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +411,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +551,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +840,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +426,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1171,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +1299,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1056,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +712,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +167,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +131,Lithuania,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +551,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1289,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1192,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1219,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +637,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +168,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +535,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1109,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1286,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +898,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +751,Lithuania,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +239,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +972,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +986,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1538,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +972,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +898,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1158,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1273,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +675,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1487,Portugal,476,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +177,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1146,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1179,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +1284,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1487,Portugal,1174,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +1080,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +854,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1244,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1252,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +955,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,722,0,Basic,Integration,Integration Techniques,Integration by parts +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +840,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1095,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,472,0,Advanced,Graph Theory,Graph Theory,Simple graph +576,Ireland,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +37,Portugal,1080,0,Basic,Differentiation,Partial Differentiation,First order +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,892,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1126,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +174,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +913,Ireland,1160,1,Basic,Set Theory,Set Theory,Subset +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +1308,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +420,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +695,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +984,Romania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1269,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1043,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1564,Portugal,682,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +26,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1427,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1081,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +28,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +509,Portugal,604,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +876,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +750,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +176,Italy,730,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,1246,1,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1274,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +599,Italy,619,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Characteristic polynomial" +643,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +551,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +634,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1032,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +725,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1146,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +750,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +576,Ireland,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +166,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +974,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +840,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1428,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +26,Portugal,831,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1324,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1079,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +669,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1284,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +175,Italy,1489,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +85,Lithuania,1150,0,Basic,Set Theory,Set Theory,Subset +26,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1077,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +625,Russian Federation,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1334,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +380,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +180,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +176,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +421,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +634,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1352,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1153,1,Basic,Set Theory,Set Theory,Subset +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +71,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +776,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +700,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +167,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1070,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +37,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +411,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1275,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +509,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +955,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +750,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +378,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1082,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +955,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +967,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +969,Romania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +200,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +155,Spain,172,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1130,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +965,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +189,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1291,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +344,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +71,Portugal,1220,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Power rule" +1322,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1149,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +634,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +168,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +636,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1564,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1252,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1271,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +470,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1132,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +37,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +177,Italy,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1292,Italy,606,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +1350,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1509,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +525,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1080,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1292,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +152,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +876,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +483,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1299,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1081,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +240,Lithuania,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +26,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +524,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +626,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +81,Lithuania,581,1,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +720,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +645,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +470,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1427,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +403,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1319,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +72,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1427,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1305,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1081,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1283,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +81,Lithuania,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +81,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1375,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1444,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +576,Ireland,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +626,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +69,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1049,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +634,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +72,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1195,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +866,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1080,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +776,Portugal,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1284,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +592,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1095,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +67,Portugal,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +509,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1332,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1338,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1049,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1023,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +28,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +511,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +979,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +375,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +524,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +71,Portugal,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +80,Lithuania,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1143,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +1243,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +36,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +509,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +890,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +90,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +599,Italy,854,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1332,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1518,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +176,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +1288,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1427,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1043,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +852,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1402,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1481,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +239,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +972,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1271,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +394,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +854,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,694,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +37,Portugal,1228,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +884,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +406,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +71,Portugal,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +637,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +591,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,121,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +967,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +979,Portugal,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +176,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +72,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1149,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +660,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1044,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +1147,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +451,Portugal,314,1,Basic,Probability,Probability,"Event,Sample space" +1542,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1150,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +470,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +349,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1047,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +853,Italy,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +91,Lithuania,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +876,Italy,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +349,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +510,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +642,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1322,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +175,Italy,1475,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1283,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +626,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +473,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +464,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +638,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1486,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1311,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1557,Portugal,86,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1122,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +409,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1173,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +426,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1091,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1375,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1256,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1157,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1252,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1252,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1557,Portugal,670,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +208,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +80,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1353,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +128,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1565,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +855,Italy,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1248,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1505,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +178,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +378,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +451,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1044,Lithuania,1001,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +359,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +57,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +52,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1271,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +974,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +81,Lithuania,935,1,Basic,Graph Theory,Graph Theory,Chromatic number +414,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +411,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +80,Lithuania,530,0,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1481,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +890,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,890,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1274,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1049,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1109,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +201,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +344,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +840,Italy,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1150,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +679,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +378,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +411,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +411,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +168,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +986,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1286,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1379,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1319,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +751,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +1525,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +406,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +509,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +71,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1286,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1173,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +176,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1292,Italy,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +852,Italy,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +510,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1256,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +151,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1352,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1171,Portugal,863,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +380,Portugal,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +582,Lithuania,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +898,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +776,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +682,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +466,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +200,Lithuania,397,1,Basic,Integration,Integration Techniques,Substitution +657,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +853,Italy,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1243,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1185,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1056,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1135,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,303,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +455,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +713,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +465,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1079,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1011,Portugal,658,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +598,Italy,249,0,Advanced,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system,Inconsistent system" +1350,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +466,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +890,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +175,Italy,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1552,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1109,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +378,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +591,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,393,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1281,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +202,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +690,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1379,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +876,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1279,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +913,Ireland,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +128,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +668,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1557,Portugal,677,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1350,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +675,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +967,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +972,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +376,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +69,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1140,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,906,0,Basic,Differential Equations,Differential Equations,Linear differential equation +731,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1279,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +840,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +966,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +455,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +218,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +511,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1171,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,168,0,Basic,Differential Equations,Differential Equations,Linear differential equation +28,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1244,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +660,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +955,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1289,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +582,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +626,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1331,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1045,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +178,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +189,Portugal,369,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +1288,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +156,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +714,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1271,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1281,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,984,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +80,Lithuania,1020,1,Basic,Optimization,Linear Optimization,"Linear programming,Assignment problem,Excel solver add-in,Simplex method" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1256,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1319,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +956,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1290,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +955,Lithuania,1086,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +552,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +41,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1244,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +979,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +525,Portugal,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +380,Portugal,1010,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +1011,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,955,0,Basic,Optimization,Linear Optimization,Linear programming +393,Lithuania,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +311,Ireland,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +955,Lithuania,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +753,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +344,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1299,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1308,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +726,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1049,Lithuania,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +175,Italy,1496,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +511,Portugal,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1082,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +853,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1173,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +855,Italy,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +71,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1087,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +885,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1282,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +741,Ireland,933,0,Basic,Graph Theory,Graph Theory,Planar graph +359,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +387,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1277,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +913,Ireland,1151,1,Basic,Set Theory,Set Theory,Subset +483,Portugal,877,1,Basic,Differentiation,Partial Differentiation,Second order +1292,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +175,Italy,1485,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +378,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1292,Italy,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +466,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +128,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1243,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +969,Romania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +405,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +899,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +955,Lithuania,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +899,Lithuania,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +1525,Portugal,281,0,Basic,Optimization,Linear Optimization,Linear programming +965,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +466,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +240,Lithuania,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +890,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +85,Lithuania,1147,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1252,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +352,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +511,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1518,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1517,Portugal,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +351,Portugal,145,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +394,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +625,Russian Federation,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +913,Ireland,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +509,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +351,Portugal,101,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule,Exponential rule" +884,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1229,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +973,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1340,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +986,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1148,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1029,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +457,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1324,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +643,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +890,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +204,Lithuania,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +898,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +525,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1511,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +525,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +1093,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +551,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +840,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +660,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +198,Italy,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1311,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +664,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1444,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +41,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1087,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +988,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +582,Lithuania,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1274,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +769,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +394,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1322,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +552,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +36,Portugal,1163,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +972,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +643,Italy,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +576,Ireland,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +407,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +176,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1029,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +403,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +633,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,662,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +483,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +81,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1518,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +41,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1091,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +973,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1308,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +625,Russian Federation,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,303,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +131,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +359,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +972,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1275,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +973,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1143,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +958,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +374,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +71,Portugal,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +851,Italy,1208,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +591,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1269,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +422,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +876,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +852,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1281,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +375,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +483,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1312,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +625,Russian Federation,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +884,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +376,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +967,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +420,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +204,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +967,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,528,1,Basic,Integration,Double Integration,"Changing order of integration,Cartesian coordinates" +26,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +394,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +176,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +407,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +37,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1194,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +197,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1075,Portugal,873,0,Basic,Differentiation,Partial Differentiation,First order +626,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +351,Portugal,484,0,Advanced,Differentiation,Derivatives,"Power rule,Product rule" +699,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +351,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +712,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +525,Portugal,512,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1173,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +483,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +165,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +582,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1071,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +374,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1036,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +853,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +61,Portugal,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +415,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +986,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1533,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +36,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1428,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +71,Portugal,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +634,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +376,Portugal,1054,1,Advanced,Complex Numbers,Complex Numbers,"Equations involving complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Real part" +1288,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +511,Portugal,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +986,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +731,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1075,Portugal,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +511,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +457,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +403,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +973,Portugal,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +898,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +85,Lithuania,1142,1,Basic,Set Theory,Set Theory,Subset +853,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +657,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +511,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1166,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1076,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +958,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1334,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1543,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1286,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1256,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1185,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +913,Ireland,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1290,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +902,Lithuania,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +464,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +174,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +638,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1428,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +965,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +311,Ireland,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +359,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +407,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +351,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +913,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +80,Lithuania,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1481,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +958,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +633,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1253,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1289,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1244,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +349,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +37,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1289,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +85,Lithuania,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +374,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,648,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +956,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1156,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +181,Italy,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +884,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +80,Lithuania,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +415,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,695,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +414,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +176,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1150,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +349,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1039,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1324,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +414,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1273,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +28,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +675,Portugal,80,1,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +955,Lithuania,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +509,Portugal,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1525,Portugal,318,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1047,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1093,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +415,Portugal,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +913,Ireland,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,1147,1,Basic,Set Theory,Set Theory,Subset +349,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +218,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +1322,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1283,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +969,Romania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +131,Lithuania,479,1,Advanced,Graph Theory,Graph Theory,"Tree,Simple graph,Spanning tree,Path" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +750,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +128,Lithuania,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +510,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +340,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1087,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1185,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1486,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +576,Ireland,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +1244,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1256,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +240,Lithuania,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +633,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +641,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1565,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +972,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +634,Italy,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +599,Italy,621,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1275,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1505,Portugal,299,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +751,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +239,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +71,Portugal,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +414,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +37,Portugal,1226,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +426,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,1489,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part" +1191,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +52,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1322,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +974,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1564,Portugal,982,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +176,Italy,1226,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Exponential rule" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,489,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1087,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +972,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1022,Lithuania,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +750,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +890,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1163,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +376,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1039,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1219,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1517,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +415,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +71,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +411,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +37,Portugal,467,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1557,Portugal,674,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +201,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +409,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1079,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +380,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +451,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +1082,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +953,Lithuania,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1147,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +643,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1080,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +988,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +26,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +576,Ireland,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +552,Portugal,159,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +349,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1284,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1049,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +913,Ireland,305,0,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +407,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +394,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1334,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1363,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,873,1,Basic,Differentiation,Partial Differentiation,First order +985,Romania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +509,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +457,Portugal,350,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1143,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1157,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1564,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +81,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +181,Italy,84,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Indeterminate forms,Limits" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1080,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1475,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number,Modulus of a complex number" +1043,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1093,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +876,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +511,Portugal,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1256,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +510,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,468,0,Basic,Graph Theory,Graph Theory,Simple graph +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1564,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1024,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1283,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +455,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1487,Portugal,936,1,Basic,Graph Theory,Graph Theory,Chromatic number +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1109,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1277,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +41,Portugal,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +406,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1411,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +885,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +548,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +510,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1332,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +90,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +884,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +582,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +199,Lithuania,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1126,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +725,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +41,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +876,Italy,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1334,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1509,Portugal,981,1,Basic,Real Functions of a single variable,Limits and Continuity,Limits +378,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +174,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +750,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1481,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +90,Lithuania,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1275,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1195,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1274,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +699,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +41,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1332,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +359,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +468,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +552,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1091,Lithuania,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1333,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1219,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +378,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +90,Lithuania,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +37,Portugal,880,1,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1077,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1081,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +394,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,502,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +966,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +380,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +1109,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1272,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +890,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,260,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +751,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1093,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1195,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1324,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1126,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1148,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +455,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +422,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +853,Italy,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +853,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +913,Ireland,1125,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +455,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +128,Lithuania,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1194,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +415,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +712,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +26,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +455,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +376,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1271,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +524,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +378,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1157,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1219,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +1322,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +175,Italy,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +344,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +509,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +451,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +988,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +664,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1324,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +71,Portugal,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1284,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +466,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1565,Portugal,85,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +250,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,475,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +855,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1146,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +956,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +71,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +899,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1557,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +985,Romania,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +551,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +374,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1271,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1300,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1157,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1350,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +683,Ireland,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1097,1,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +855,Italy,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +219,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +202,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1564,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1158,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1043,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +913,Ireland,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +181,Italy,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +422,Portugal,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1291,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,252,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +483,Portugal,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1252,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1093,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +81,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1557,Portugal,408,1,Advanced,Integration,Integration Techniques,Integration by parts +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +37,Portugal,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +645,Italy,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1272,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +37,Portugal,866,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +177,Italy,98,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1219,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +394,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +200,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +955,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1273,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +72,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +403,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +176,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1049,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +175,Italy,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +853,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +174,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1334,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1079,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +511,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +378,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +91,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +36,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1195,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1311,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +64,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +598,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +748,Portugal,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +751,Lithuania,717,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1525,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +725,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +852,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,1486,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +483,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +380,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +406,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +582,Lithuania,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +199,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1173,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1271,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +684,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +633,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +41,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +725,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,157,0,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +468,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1289,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +626,Italy,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +854,Italy,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +552,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +899,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +689,Portugal,820,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1109,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +178,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +71,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +899,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1135,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1487,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +625,Russian Federation,663,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1288,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +918,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1321,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,999,1,Basic,Differentiation,Partial Differentiation,First order +1034,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +955,Lithuania,1111,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1197,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +510,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +421,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1542,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,746,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +633,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +176,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1276,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +128,Lithuania,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1171,Portugal,1015,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +411,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +986,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1322,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +953,Lithuania,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +884,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +41,Portugal,387,0,Basic,Graph Theory,Graph Theory,"Shortest path,Tree,Spanning tree,Path" +178,Italy,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +394,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +37,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +359,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,520,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +511,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +510,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1402,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +166,Portugal,546,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +576,Ireland,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1080,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +483,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1322,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1173,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +979,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +524,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +426,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +85,Lithuania,362,0,Basic,Graph Theory,Graph Theory,Simple graph +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +374,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +349,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +406,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1324,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1274,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,617,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +634,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1149,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1256,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +409,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1289,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +380,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1082,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1219,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1056,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +689,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +176,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +41,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +1197,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1049,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +974,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +81,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +26,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +985,Romania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +415,Portugal,140,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Inverse matrix" +406,Portugal,651,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1075,Portugal,1002,1,Basic,Differentiation,Partial Differentiation,First order +1288,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1561,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +175,Italy,1481,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +524,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +415,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,245,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +189,Portugal,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1256,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1023,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +840,Italy,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1312,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +250,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +986,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1561,Portugal,408,0,Advanced,Integration,Integration Techniques,Integration by parts +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1203,0,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1538,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +552,Portugal,677,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1079,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1351,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1292,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +598,Italy,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +155,Spain,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1135,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +128,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,674,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +72,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +918,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1481,Portugal,465,0,Basic,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +168,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +876,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +885,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +651,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +972,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +52,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +473,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +890,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +71,Portugal,314,0,Basic,Probability,Probability,"Event,Sample space" +1182,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,1,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +1363,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +551,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +470,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1230,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +36,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +642,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +852,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +525,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +465,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +28,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +509,Portugal,597,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +378,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1308,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1305,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +840,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1322,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1050,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +374,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +510,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +202,Lithuania,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +633,Italy,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +641,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +174,Portugal,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +913,Ireland,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +625,Russian Federation,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +576,Ireland,574,1,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +599,Italy,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +376,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +638,Italy,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1402,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +955,Lithuania,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1564,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1411,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1150,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1079,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +509,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +374,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +535,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +689,Portugal,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +80,Lithuania,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +855,Italy,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1402,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +378,Portugal,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +473,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +598,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +81,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1036,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +510,Portugal,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1289,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1351,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1286,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +852,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1071,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +175,Italy,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1284,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,592,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +250,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +359,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +41,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +72,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +81,Lithuania,473,1,Advanced,Graph Theory,Graph Theory,Simple graph +142,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +473,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1283,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +128,Lithuania,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1077,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +775,Portugal,761,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +414,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +167,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +239,Lithuania,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1191,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,858,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1487,Portugal,1207,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1276,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +548,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +854,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1334,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +776,Portugal,550,1,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +397,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +69,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +576,Ireland,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +36,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1095,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1230,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +966,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +393,Lithuania,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +151,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1291,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +208,Lithuania,129,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +510,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +311,Ireland,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +509,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +344,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1308,Italy,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +349,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +90,Lithuania,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1281,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +155,Spain,146,0,Basic,Differential Equations,Differential Equations,Separable variables equation +890,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +91,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1353,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +80,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +979,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +91,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +638,Italy,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1518,Portugal,134,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +988,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1375,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +420,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +28,Portugal,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +426,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1173,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +36,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1163,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,1000,1,Basic,Differentiation,Partial Differentiation,First order +509,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1032,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +483,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +36,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,510,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1091,Lithuania,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +464,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +359,Portugal,505,0,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +85,Lithuania,1139,0,Basic,Set Theory,Set Theory,Subset +913,Ireland,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1109,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +26,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +177,Italy,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +57,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +955,Lithuania,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +582,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +380,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +751,Lithuania,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +455,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +422,Portugal,124,1,Advanced,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Inverse matrix ,Hermitian matrix,Commuting matrices,Linearly independent rows,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon,Determinant computation,Matrix multiplication,Determinant properties" +71,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1044,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +918,Portugal,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +966,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +473,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1351,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1126,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +918,Portugal,1000,0,Basic,Differentiation,Partial Differentiation,First order +1324,Italy,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +393,Lithuania,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1332,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,627,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +483,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +860,Italy,234,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +511,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +876,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1130,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +979,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +664,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +551,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +645,Italy,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1284,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +775,Portugal,488,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +420,Portugal,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +80,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +200,Lithuania,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1163,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +198,Italy,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +576,Ireland,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +406,Portugal,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1402,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,792,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1081,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1081,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +525,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +85,Lithuania,475,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +1146,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1505,Portugal,305,1,Basic,Probability,Probability,"Event,Axioms of probability,Addition, multiplication, and total probability rules,Sample space" +1411,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1284,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +344,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +552,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +509,Portugal,90,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +890,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +633,Italy,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +647,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1516,Portugal,956,1,Basic,Optimization,Linear Optimization,Linear programming +1171,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +411,Portugal,724,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +525,Portugal,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +37,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +376,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +511,Portugal,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1248,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +663,Portugal,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +972,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1147,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,943,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +985,Romania,789,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1486,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1291,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +311,Ireland,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +167,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +168,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +1339,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +600,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +511,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1076,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1557,Portugal,832,0,Basic,Differential Equations,Differential Equations,Separable variables equation +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1179,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +465,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1256,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +378,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1091,Lithuania,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1289,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +80,Lithuania,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1352,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +525,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +380,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +90,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +972,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1185,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +448,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +167,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1324,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +876,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +375,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +375,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +535,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +552,Portugal,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1427,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +155,Spain,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1196,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1352,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1273,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +692,Portugal,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1517,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1557,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1279,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1135,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,530,1,Basic,Integration,Double Integration,"Iterate integrals,Area ,y-simple region (type I),Cartesian coordinates" +37,Portugal,595,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +37,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +376,Portugal,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1028,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +411,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +966,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +375,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +201,Lithuania,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +985,Romania,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1379,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +637,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1163,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1283,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,1227,0,Basic,Differentiation,Derivatives,Exponential rule +349,Portugal,753,0,Basic,Differentiation,Derivatives,Exponential rule +509,Portugal,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1146,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +168,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +81,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +956,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1039,Lithuania,879,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +854,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,529,0,Basic,Integration,Double Integration,Iterate integrals +1241,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1109,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +852,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1082,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1158,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +457,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +52,Portugal,511,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +37,Portugal,661,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +85,Lithuania,1132,0,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,887,1,Advanced,Differentiation,Partial Differentiation,First order +1252,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +979,Portugal,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +455,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1339,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1130,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +741,Ireland,947,1,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +985,Romania,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +690,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +466,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1135,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1044,Lithuania,176,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1516,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +155,Spain,903,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1379,Portugal,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1428,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +250,Lithuania,134,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1149,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1509,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1486,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +202,Lithuania,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1543,Portugal,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +918,Portugal,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +52,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +860,Italy,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +509,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1341,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +898,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +168,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +351,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +524,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1147,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1289,Italy,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +913,Ireland,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1444,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +955,Lithuania,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +71,Portugal,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1077,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +165,Portugal,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1076,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +750,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +174,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +636,Italy,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +1538,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +26,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +178,Italy,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +80,Lithuania,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1179,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +626,Italy,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1351,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +967,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1256,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +876,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1274,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +64,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +473,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +411,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1080,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +72,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1093,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +1150,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +854,Italy,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1322,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1029,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1093,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +181,Italy,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +985,Romania,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1517,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +510,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +167,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +666,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +966,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +181,Italy,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +166,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +181,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +470,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1341,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1081,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1312,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +375,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +448,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +626,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1093,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +142,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +913,Ireland,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1157,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1135,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +420,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +509,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +692,Portugal,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +142,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +90,Lithuania,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +967,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +599,Italy,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +689,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +751,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +524,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +966,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +90,Lithuania,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +667,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +394,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1173,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +748,Portugal,480,0,Basic,Graph Theory,Graph Theory,Planar graph +972,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +776,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +634,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1271,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +885,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1511,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1132,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1288,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +178,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1269,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,405,1,Advanced,Integration,Integration Techniques,Rational functions +1024,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +26,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +26,Portugal,835,1,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +510,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +378,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1103,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +769,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1565,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1564,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +851,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +986,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +988,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +525,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1158,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +37,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1085,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1324,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +966,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +956,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1146,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +26,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +899,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1291,Italy,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1334,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +349,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +81,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +403,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,888,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1253,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1140,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1156,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +591,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +966,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +969,Romania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1286,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1157,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +592,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1032,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +163,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +853,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1333,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1505,Portugal,959,1,Basic,Optimization,Linear Optimization,Linear programming +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1483,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +884,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +37,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1158,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +890,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +176,Italy,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +359,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,1003,0,Basic,Optimization,Linear Optimization,Linear programming +1024,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +351,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +181,Italy,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +986,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +64,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1082,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +175,Italy,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1179,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1044,Lithuania,912,1,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +419,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1402,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +592,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +855,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +633,Italy,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +467,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1289,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +422,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +448,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1029,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1481,Portugal,553,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +1322,Italy,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1505,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +203,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +163,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1150,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +41,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +406,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +956,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1179,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +511,Portugal,216,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +884,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1290,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +420,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +966,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1219,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +167,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1340,Portugal,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +552,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +204,Lithuania,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +28,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1079,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1288,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +899,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +483,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1148,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +351,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1311,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +374,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1274,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1542,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +394,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +380,Portugal,1004,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +174,Portugal,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1082,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,741,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +426,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1420,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +36,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1290,Italy,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1289,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +899,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1557,Portugal,675,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1352,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1076,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +1082,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1146,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +189,Portugal,512,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +972,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +955,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +898,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +61,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1274,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1026,Lithuania,269,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +181,Italy,922,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +965,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +239,Lithuania,95,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +403,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +776,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1324,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +884,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +876,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +966,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +851,Italy,1005,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +972,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +394,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +387,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +26,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +591,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +775,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1076,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1022,Lithuania,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +91,Lithuania,311,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +177,Italy,734,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +750,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1444,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +576,Ireland,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1274,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +731,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1321,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1076,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +175,Italy,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +854,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1322,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +1481,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1517,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1049,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +455,Portugal,122,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1516,Portugal,164,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1195,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1143,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +753,Italy,998,0,Basic,Differentiation,Partial Differentiation,First order +1271,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1256,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1557,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +890,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,1128,1,Basic,Set Theory,Set Theory,Subset +851,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +657,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1289,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1109,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +26,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1109,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +37,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1149,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +374,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1363,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,1146,1,Basic,Set Theory,Set Theory,Subset +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +174,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1525,Portugal,760,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +37,Portugal,963,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1179,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1289,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +176,Italy,721,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +1087,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1081,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1311,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1557,Portugal,684,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1338,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1291,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1272,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +239,Lithuania,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +899,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1076,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +455,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +163,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1022,Lithuania,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1244,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +633,Italy,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +176,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1291,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +913,Ireland,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1219,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1149,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +646,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +913,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +465,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +876,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +380,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1256,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +741,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +71,Portugal,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +218,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1505,Portugal,744,1,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1271,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +394,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1339,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1517,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +483,Portugal,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +198,Italy,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +899,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +411,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +85,Lithuania,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +985,Romania,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +464,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,282,1,Basic,Optimization,Linear Optimization,Linear programming +1283,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +26,Portugal,1162,1,Basic,Set Theory,Set Theory,Subset +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +524,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +80,Lithuania,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1080,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +720,Portugal,186,1,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +969,Romania,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +64,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +898,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1487,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +81,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +175,Italy,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1428,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1024,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +414,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,524,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +576,Ireland,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +750,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +407,Portugal,496,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1564,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +827,Italy,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +80,Lithuania,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +876,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1252,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +967,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +899,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +636,Italy,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1322,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1192,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +422,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +351,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +359,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +465,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +860,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +203,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1334,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +913,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +37,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1557,Portugal,688,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +380,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +483,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +884,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1291,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,186,1,Basic,Statistics,Statistics,"Population,Sample" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1505,Portugal,957,1,Basic,Optimization,Linear Optimization,Linear programming +973,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1149,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +466,Portugal,348,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1077,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1148,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1332,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1525,Portugal,747,0,Advanced,Optimization,Linear Optimization,"Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis,Postoptimality analysis" +913,Ireland,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +175,Italy,1061,0,Basic,Complex Numbers,Complex Numbers,Complex plane +26,Portugal,1149,0,Basic,Set Theory,Set Theory,Subset +955,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +511,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +85,Lithuania,1163,1,Basic,Set Theory,Set Theory,Subset +37,Portugal,406,1,Basic,Integration,Integration Techniques,Direct inspection +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1077,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +1273,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1561,Portugal,403,0,Advanced,Integration,Integration Techniques,Rational functions +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +41,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1244,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1509,Portugal,682,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +91,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1269,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +972,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1179,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +913,Ireland,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +380,Portugal,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +840,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +955,Lithuania,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1252,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +827,Italy,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +1071,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +71,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,861,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1256,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +465,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +851,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1147,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +200,Lithuania,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +667,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +955,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +969,Romania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +510,Portugal,217,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1486,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +969,Romania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +660,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +420,Portugal,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +600,Italy,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +509,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +378,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1339,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +918,Portugal,95,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule,Exponential rule" +466,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1147,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +204,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1036,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1289,Italy,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1339,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +775,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +913,Ireland,952,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1274,Portugal,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +177,Italy,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1273,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1282,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +167,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1229,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1538,Portugal,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +85,Lithuania,1124,0,Basic,Set Theory,Set Theory,"Integration by parts,Substitution,Simple integration,Direct inspection" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1179,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +647,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1146,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1379,Portugal,611,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +690,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +953,Lithuania,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +359,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1043,Lithuania,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +80,Lithuania,493,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +854,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,551,1,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +624,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +988,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +625,Russian Federation,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +387,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1197,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1029,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +898,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1516,Portugal,903,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1322,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1229,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1275,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +855,Italy,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +393,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,559,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +426,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +311,Ireland,639,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +511,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +455,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1157,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1157,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +344,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +41,Portugal,159,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Integral curve,Cauchy problem" +854,Italy,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1322,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +700,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +411,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +426,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +414,Portugal,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1219,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +71,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +80,Lithuania,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +239,Lithuania,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +955,Lithuania,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +151,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +67,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1319,Italy,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +955,Lithuania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +403,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +769,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1275,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1542,Portugal,1102,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +37,Portugal,604,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +955,Lithuania,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +376,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1286,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1271,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1252,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1093,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +855,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +913,Ireland,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1402,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1279,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +898,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1140,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +965,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1479,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +625,Russian Federation,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +57,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1552,Portugal,719,0,Basic,Integration,Integration Techniques,Integration by parts +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +26,Portugal,828,1,Basic,Differential Equations,Differential Equations,Linear differential equation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +524,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1135,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +378,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1130,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +675,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +576,Ireland,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +986,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +81,Lithuania,362,1,Basic,Graph Theory,Graph Theory,Simple graph +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +41,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1147,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1247,Portugal,642,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +851,Italy,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +406,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1542,Portugal,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +969,Romania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +387,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +647,Ireland,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +483,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,255,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +955,Lithuania,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1039,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +955,Lithuania,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +884,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1308,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +406,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1286,Italy,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +28,Portugal,700,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +851,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,1151,0,Basic,Set Theory,Set Theory,Subset +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1173,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,624,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1308,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +651,Ireland,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +990,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +509,Portugal,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +483,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1564,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +168,Portugal,392,1,Advanced,Integration,Integration Techniques,Substitution +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1411,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +167,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1352,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +72,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1352,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,491,0,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +374,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +510,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +979,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +693,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +36,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,620,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +168,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +415,Portugal,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1150,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +979,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +359,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +600,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1185,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +344,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1379,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1516,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1196,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1481,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +1292,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +406,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +548,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1517,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +151,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +509,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +913,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1241,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1229,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +466,Portugal,359,0,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1322,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,224,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +973,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1241,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1479,Portugal,1208,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +387,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +36,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1352,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1525,Portugal,743,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +1427,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,1229,0,Basic,Differentiation,Partial Differentiation,First order +41,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1511,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +852,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1312,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1525,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +483,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +966,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +750,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +666,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +91,Lithuania,714,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +80,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +71,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +1087,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +972,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +860,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1037,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,1072,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +448,Portugal,491,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +982,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1286,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +851,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +483,Portugal,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +913,Ireland,316,0,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +69,Portugal,566,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +852,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1219,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +661,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +465,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1487,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +420,Portugal,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,622,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +679,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +420,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +634,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +598,Italy,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +840,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1564,Portugal,84,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +967,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1564,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1146,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +1130,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +885,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +71,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +37,Portugal,1033,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +351,Portugal,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +693,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +509,Portugal,125,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +885,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1077,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1095,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +28,Portugal,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,309,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +403,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +80,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +155,Spain,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +457,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1504,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +689,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +972,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +81,Lithuania,481,0,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +683,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +851,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +552,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +884,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1487,Portugal,720,1,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +599,Italy,623,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +851,Italy,818,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +41,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +884,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +168,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1526,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Linear equations,Quadratic equations,Simplify expressions" +1087,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +175,Italy,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +71,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1109,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +669,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +351,Portugal,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +80,Lithuania,514,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +80,Lithuania,280,1,Basic,Optimization,Linear Optimization,Linear programming +1557,Portugal,684,1,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1308,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,887,1,Advanced,Differentiation,Partial Differentiation,First order +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +658,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1146,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1525,Portugal,767,0,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +387,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +28,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +407,Portugal,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +725,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1022,Lithuania,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1146,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +468,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1081,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +552,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +41,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1241,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +67,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +174,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1525,Portugal,750,0,Advanced,Optimization,Linear Optimization,"Sensitivity analysis,Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming" +376,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +457,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +26,Portugal,829,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1279,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +972,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1229,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +37,Portugal,872,0,Basic,Differential Equations,Differential Equations,Linear differential equation +374,Portugal,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +378,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +380,Portugal,1015,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +885,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +899,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +576,Ireland,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1203,1,Basic,Integration,Definite Integrals,"Volume of revolution,Fundamental theorem of Calculus" +1087,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +664,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1132,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1505,Portugal,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +956,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +955,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1542,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1375,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +28,Portugal,547,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +955,Lithuania,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1538,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1281,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1284,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +769,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1517,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1047,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +155,Spain,96,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +71,Portugal,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +890,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +175,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +37,Portugal,1073,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +982,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1256,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +511,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +468,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1269,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +509,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1275,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +986,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +26,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +397,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1195,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +41,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +411,Portugal,723,0,Basic,Integration,Integration Techniques,Substitution +80,Lithuania,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +674,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +69,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1194,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1077,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +359,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1196,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1158,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +168,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +955,Lithuania,1214,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +81,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +633,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +552,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1504,Portugal,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +176,Italy,729,0,Basic,Differentiation,Derivatives,Quotient rule +1565,Portugal,732,0,Advanced,Integration,Integration Techniques,Trigonometric substitution +26,Portugal,829,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1299,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1150,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1215,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +85,Lithuania,477,0,Basic,Graph Theory,Graph Theory,Simple graph +1487,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1351,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1197,Slovenia,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +625,Russian Federation,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,555,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1271,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1171,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1095,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +411,Portugal,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +91,Lithuania,460,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +988,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +208,Lithuania,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,848,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1339,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +178,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1286,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +85,Lithuania,476,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +986,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1339,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +218,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +64,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1479,Portugal,1053,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1565,Portugal,905,1,Advanced,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +695,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +349,Portugal,144,1,Basic,Differentiation,Derivatives,Trigonometric rules +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +451,Portugal,490,1,Advanced,Probability,Probability,"Addition, multiplication, and total probability rules,Independence,Event" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +403,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +634,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +340,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1351,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +625,Russian Federation,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +483,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +975,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +128,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +1557,Portugal,87,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +750,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +359,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1487,Portugal,929,1,Basic,Graph Theory,Graph Theory,"Cycle and circuit,Shortest path,Path,Planar graph,Tree,Adjacency matrix,Chromatic number,Simple graph,Spanning tree,Incidence matrix" +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +69,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1247,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +509,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1340,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +202,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1334,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1147,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +918,Portugal,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +420,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +913,Ireland,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1091,Lithuania,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +913,Ireland,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +155,Spain,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +509,Portugal,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +219,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1220,Slovenia,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1300,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +168,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +26,Portugal,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +1135,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,623,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +1081,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1146,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,738,1,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +28,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1044,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +690,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1070,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1487,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +751,Lithuania,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +483,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1196,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +700,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +750,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1289,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +674,Ireland,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +71,Portugal,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +854,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +851,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +85,Lithuania,1127,0,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1195,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1288,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +876,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1308,Italy,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1248,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +913,Ireland,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +176,Italy,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +956,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +28,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1179,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1081,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1308,Italy,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +695,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +91,Lithuania,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +483,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +599,Italy,617,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Eigenvector,Characteristic polynomial" +352,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1517,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +483,Portugal,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1247,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1150,0,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +406,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +470,Portugal,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1091,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,837,1,Basic,Differential Equations,Differential Equations,Linear differential equation +250,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1135,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,569,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +840,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1079,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +81,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1252,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +525,Portugal,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +985,Romania,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +151,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1081,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +855,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +165,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1056,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +37,Portugal,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1273,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1331,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +394,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1038,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +899,Lithuania,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +827,Italy,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +91,Lithuania,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1292,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +414,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1292,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1487,Portugal,1052,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1350,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +509,Portugal,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1182,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +552,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +72,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +468,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +406,Portugal,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +37,Portugal,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +426,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +426,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +985,Romania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,412,1,Basic,Integration,Integration Techniques,Power of trigonometric functions +851,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +397,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1182,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +151,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +204,Lithuania,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,767,0,Basic,Probability,Probability,"Event,Sample space" +1284,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1219,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +168,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1082,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1026,Lithuania,263,0,Basic,Differentiation,Partial Differentiation,First order +181,Italy,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1288,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +352,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1428,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +90,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +967,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +667,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1275,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +411,Portugal,733,0,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +28,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1291,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +851,Italy,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +411,Portugal,610,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +374,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +85,Lithuania,1144,1,Basic,Set Theory,Set Theory,Subset +576,Ireland,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +72,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +61,Portugal,156,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1543,Portugal,1087,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +899,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +67,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1353,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +973,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1244,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1135,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +168,Portugal,388,0,Advanced,Integration,Integration Techniques,Rational functions +1185,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +974,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1080,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +535,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +85,Lithuania,1151,1,Basic,Set Theory,Set Theory,Subset +1308,Italy,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +165,Portugal,542,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Area ,Iterate integrals" +638,Italy,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +973,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +37,Portugal,1173,1,Basic,Integration,Definite Integrals,"Integration by parts,Fundamental theorem of Calculus" +467,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +155,Spain,493,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangles,Triangle inequality" +1428,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +769,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +840,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1148,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1487,Portugal,145,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1289,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +725,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +81,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +509,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1481,Portugal,671,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1444,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1338,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1126,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1130,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +1411,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +598,Italy,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +1081,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1163,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +576,Ireland,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1517,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +91,Lithuania,456,1,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1171,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +851,Italy,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +699,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1319,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,1070,1,Basic,Differentiation,Partial Differentiation,First order +646,Ireland,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +197,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,592,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Logarithmic rule,Product rule,Trigonometric rules" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +854,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1276,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1050,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +394,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +359,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +52,Portugal,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +1049,Lithuania,260,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1321,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +958,Lithuania,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1269,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1071,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +41,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1444,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,643,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +625,Russian Federation,662,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +468,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +71,Portugal,650,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +637,Italy,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +465,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +969,Romania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1146,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +483,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +176,Italy,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +26,Portugal,146,1,Basic,Differential Equations,Differential Equations,Separable variables equation +37,Portugal,1014,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1291,Italy,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +969,Romania,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1044,Lithuania,148,1,Basic,Differentiation,Partial Differentiation,First order +67,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +374,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1271,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +352,Portugal,100,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1135,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1219,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1022,Lithuania,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +37,Portugal,1168,0,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +626,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +639,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +168,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1517,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +168,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1334,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1082,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +142,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1173,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +972,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +203,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +664,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +1427,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +457,Portugal,330,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +986,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +80,Lithuania,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1049,Lithuania,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +854,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +36,Portugal,1151,1,Basic,Set Theory,Set Theory,Subset +855,Italy,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +37,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +524,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +827,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1276,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1322,Italy,454,1,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +974,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +178,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1334,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,1138,1,Basic,Set Theory,Set Theory,Subset +1044,Lithuania,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +90,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +510,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1505,Portugal,488,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1338,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +422,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +176,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +625,Russian Federation,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1564,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +1487,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +625,Russian Federation,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,1067,0,Advanced,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +576,Ireland,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +988,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +175,Italy,1496,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +625,Russian Federation,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,866,1,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Optimal solution,Minimum" +155,Spain,904,0,Basic,Differential Equations,Differential Equations,"Bernoulli equation,Linear differential equation" +899,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +1039,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,634,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +473,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +876,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +898,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +200,Lithuania,733,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1179,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +175,Italy,1474,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +511,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1023,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1288,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1252,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,514,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1277,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,950,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1352,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1143,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1252,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1340,Portugal,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1196,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +128,Lithuania,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1070,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +28,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +64,Portugal,576,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +634,Italy,657,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1322,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1066,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +1289,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +90,Lithuania,133,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +175,Italy,1485,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +853,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +204,Lithuania,366,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +28,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +955,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1082,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +167,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1487,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +174,Portugal,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1322,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +420,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +972,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +1308,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1082,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +80,Lithuania,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1049,Lithuania,1002,0,Basic,Differentiation,Partial Differentiation,First order +168,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1081,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +351,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +351,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +990,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +625,Russian Federation,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +174,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1219,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1332,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +625,Russian Federation,513,0,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +967,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1517,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +483,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +884,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1282,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1351,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,494,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +741,Ireland,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1122,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +175,Italy,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +374,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +852,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +576,Ireland,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +885,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,222,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +72,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +636,Italy,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1518,Portugal,767,1,Basic,Probability,Probability,"Event,Sample space" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +387,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1087,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +840,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +524,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1277,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +853,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1538,Portugal,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +470,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1147,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1341,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +69,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,349,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +551,Lithuania,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +690,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +898,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +28,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1173,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +177,Italy,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +1291,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +71,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,958,0,Basic,Optimization,Linear Optimization,Linear programming +28,Portugal,699,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +376,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +899,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +876,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +876,Italy,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +524,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +41,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1081,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +524,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +549,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +26,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +387,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +250,Lithuania,218,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +71,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +166,Portugal,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +426,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1081,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1411,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1122,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +625,Russian Federation,799,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +775,Portugal,301,0,Basic,Probability,Probability,"Sample space,Event" +1338,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,363,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,545,0,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +1286,Italy,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +1087,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +414,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1286,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +956,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +884,Italy,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +462,Lithuania,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1082,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1565,Portugal,412,0,Basic,Integration,Integration Techniques,Power of trigonometric functions +986,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1050,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +411,Portugal,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1479,Portugal,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +955,Lithuania,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +387,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +91,Lithuania,369,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures,Classification of geometrical figures" +72,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +956,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +483,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1379,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +582,Lithuania,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +1286,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1166,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1095,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +840,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +198,Italy,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1333,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +421,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +884,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1299,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +36,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +683,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +1018,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +160,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +41,Portugal,584,1,Basic,Graph Theory,Graph Theory,Adjacency matrix +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +967,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +455,Portugal,640,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1518,Portugal,135,0,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +525,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +41,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +974,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1411,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1271,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,281,1,Basic,Optimization,Linear Optimization,Linear programming +576,Ireland,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +91,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +41,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +91,Lithuania,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1308,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +411,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +525,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +689,Portugal,964,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +674,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +175,Italy,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +884,Italy,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1281,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +658,Ireland,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1044,Lithuania,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +168,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1505,Portugal,742,0,Basic,Optimization,Linear Optimization,"Duality theory,Linear programming" +972,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +851,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +41,Portugal,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1244,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1564,Portugal,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +197,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1289,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +684,Ireland,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +660,Portugal,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +979,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +953,Lithuania,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +80,Lithuania,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +975,Portugal,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +1379,Portugal,925,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +168,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +175,Italy,1092,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +969,Romania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +840,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,458,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +753,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +393,Lithuania,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +168,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1192,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +700,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1271,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1509,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +913,Ireland,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1324,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,736,1,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1219,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1322,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +898,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1334,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +28,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +851,Italy,821,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +168,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +414,Portugal,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,802,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1157,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1288,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +509,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +349,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1158,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +576,Ireland,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +387,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1379,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +376,Portugal,355,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +899,Lithuania,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +626,Italy,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1150,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1077,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +853,Italy,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +71,Portugal,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +359,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +352,Portugal,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +1109,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +750,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +473,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +600,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1292,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +582,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +525,Portugal,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +851,Italy,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +36,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +359,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +457,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +776,Portugal,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +181,Italy,918,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1034,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1230,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +699,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +511,Portugal,92,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +884,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +376,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,565,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +91,Lithuania,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1517,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1517,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1093,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +524,Portugal,228,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +684,Ireland,182,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1213,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +1029,Lithuania,262,0,Basic,Differentiation,Partial Differentiation,First order +380,Portugal,910,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Lagrange multipliers,Saddle point,Unconstrained optimization,Second derivative test" +1543,Portugal,1089,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1171,Portugal,960,0,Basic,Optimization,Nonlinear Optimization,Linear programming +679,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1505,Portugal,962,1,Basic,Optimization,Linear Optimization,"Simplex method,Duality theory" +1070,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1080,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +661,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1277,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +85,Lithuania,1128,1,Basic,Set Theory,Set Theory,Subset +91,Lithuania,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +81,Lithuania,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1023,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1487,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1402,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1308,Italy,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +80,Lithuania,541,0,Basic,Integration,Double Integration,"y-simple region (type I),Cartesian coordinates,Iterate integrals" +1147,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +827,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1273,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,794,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1288,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1037,Lithuania,1001,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1157,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1564,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1561,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1561,Portugal,392,0,Advanced,Integration,Integration Techniques,Substitution +840,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1300,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1147,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +407,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1305,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +72,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +535,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1543,Portugal,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1196,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +854,Italy,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +64,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1334,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +913,Ireland,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +72,Portugal,548,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +1274,Portugal,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +380,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1077,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +509,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +81,Lithuania,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +168,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +965,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1244,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1275,Lithuania,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +71,Portugal,639,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Determinant computation,Determinant properties,Matrix operations" +197,Portugal,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,593,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1557,Portugal,670,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1143,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +967,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +422,Portugal,133,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +403,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1339,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +71,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1279,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1288,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1081,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +80,Lithuania,332,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +967,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +913,Ireland,1119,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +411,Portugal,596,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +955,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +551,Lithuania,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +913,Ireland,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +661,Portugal,81,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +851,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +955,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1081,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +884,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +625,Russian Federation,460,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Platonic solids,Geometric solids" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1339,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +598,Italy,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1291,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +898,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +406,Portugal,230,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1182,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +240,Lithuania,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1087,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +776,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +197,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +956,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1375,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +168,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +918,Portugal,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +753,Italy,876,0,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1286,Italy,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +483,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1049,Lithuania,269,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +876,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +166,Portugal,573,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +202,Lithuania,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1481,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +470,Portugal,554,1,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1481,Portugal,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +524,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +853,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +524,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +898,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +165,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +142,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +884,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1126,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +203,Lithuania,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +689,Portugal,862,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1517,Portugal,854,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +374,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +405,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1022,Lithuania,965,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming,Transportation problem" +511,Portugal,612,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,676,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1291,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1284,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +483,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +965,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +69,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1252,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1026,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1339,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1146,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1256,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +378,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1504,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +37,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +80,Lithuania,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +890,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +683,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +840,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +344,Portugal,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1076,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +411,Portugal,94,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1135,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +551,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +956,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +405,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +168,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +853,Italy,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1324,Italy,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1082,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1032,Lithuania,1000,0,Basic,Differentiation,Partial Differentiation,First order +898,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1135,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +634,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1505,Portugal,1021,0,Basic,Optimization,Linear Optimization,"Simplex method,Linear programming,Assignment problem,Excel solver add-in" +1182,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +200,Lithuania,723,0,Basic,Integration,Integration Techniques,Substitution +1156,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +750,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1332,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +349,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +637,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1334,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1135,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +91,Lithuania,504,0,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1322,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +91,Lithuania,498,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1093,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +854,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +239,Lithuania,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +37,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1252,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1109,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +176,Italy,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +175,Italy,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +175,Italy,1216,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +61,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +168,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +751,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +511,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +967,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +985,Romania,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1256,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1182,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +524,Portugal,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1282,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1044,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1252,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +885,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +177,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +851,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1557,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +625,Russian Federation,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1109,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,302,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +1146,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +394,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1198,0,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +1288,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1271,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1286,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1256,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +201,Lithuania,786,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1109,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +421,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1339,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +990,Portugal,625,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +176,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +52,Portugal,137,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +918,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +175,Italy,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +197,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1271,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +91,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +91,Lithuania,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +359,Portugal,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +168,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,749,0,Advanced,Optimization,Linear Optimization,"Postoptimality analysis,Duality theory,Excel solver add-in,Simplex method,Linear programming,Sensitivity analysis" +411,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +658,Ireland,184,0,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +72,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +69,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1194,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +660,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1242,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +511,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +525,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +725,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +986,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1542,Portugal,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +720,Portugal,181,1,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +1542,Portugal,1106,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +876,Italy,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +311,Ireland,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1044,Lithuania,257,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +955,Lithuania,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +988,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1091,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +57,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1427,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +965,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +201,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +890,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,744,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +955,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +840,Italy,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +717,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1247,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1024,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1322,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +406,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +599,Italy,610,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +167,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1243,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1289,Italy,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1044,Lithuania,266,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +751,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +177,Italy,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1481,Portugal,192,1,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1288,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1087,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +64,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1319,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +483,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1044,Lithuania,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +203,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +351,Portugal,660,0,Advanced,Differentiation,Derivatives,"Product rule,Trigonometric rules,Quotient rule" +1271,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1150,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +483,Portugal,503,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +669,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +473,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +885,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +359,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +375,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +473,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +483,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +986,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1518,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +953,Lithuania,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +658,Ireland,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1087,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +378,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1146,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +72,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +641,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1324,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +175,Italy,1052,0,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Algebraic form,Operations with complex numbers,Real part,Equations involving complex numbers" +551,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1247,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +41,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +422,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +988,Portugal,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +26,Portugal,1144,1,Basic,Set Theory,Set Theory,Subset +455,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +71,Portugal,909,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,559,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1291,Italy,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +175,Italy,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1402,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +876,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +473,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1079,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +683,Ireland,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1242,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +576,Ireland,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +1143,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +26,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +189,Portugal,363,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +359,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +988,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1087,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1322,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1122,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1339,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +985,Romania,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +600,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +409,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +175,Italy,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +913,Ireland,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1143,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1157,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +599,Italy,606,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Diagonalization" +80,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1026,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +470,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +91,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +37,Portugal,97,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +979,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +851,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1247,Portugal,340,0,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +511,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1311,Italy,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +851,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +890,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +198,Italy,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +918,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +91,Lithuania,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +956,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,297,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +415,Portugal,132,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +700,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +199,Lithuania,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +969,Romania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +483,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +956,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +975,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1093,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +552,Portugal,835,0,Basic,Differential Equations,Differential Equations,"Exact differential form,Cauchy problem" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +376,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1379,Portugal,920,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +168,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1557,Portugal,920,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1525,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +37,Portugal,881,1,Advanced,Differentiation,Partial Differentiation,"Directional derivative,Gradient" +890,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +393,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +955,Lithuania,1110,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1023,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,131,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +510,Portugal,497,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1284,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +177,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +128,Lithuania,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +28,Portugal,572,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +955,Lithuania,1216,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +91,Lithuania,494,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +376,Portugal,350,0,Advanced,Complex Numbers,Complex Numbers,Complex plane +913,Ireland,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +913,Ireland,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +455,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1229,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,870,0,Basic,Differential Equations,Differential Equations,Homogeneous equation +483,Portugal,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1379,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +407,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +840,Italy,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1333,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +90,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +1158,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +80,Lithuania,529,1,Basic,Integration,Double Integration,Iterate integrals +913,Ireland,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +165,Portugal,94,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1288,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1163,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +552,Portugal,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +91,Lithuania,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +91,Lithuania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +26,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +411,Portugal,629,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +693,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +91,Lithuania,456,0,Advanced,Fundamental Mathematics,Elementary Geometry,Triangles +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +625,Russian Federation,790,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1322,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1481,Portugal,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +956,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1331,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +176,Italy,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1109,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,1016,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +448,Portugal,320,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1179,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1284,Italy,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1284,Italy,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +457,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +913,Ireland,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +411,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +965,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,793,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +511,Portugal,642,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Matrix multiplication,Matrix operations" +1076,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +168,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1351,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +422,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +26,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +551,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +1182,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1411,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1182,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +374,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +415,Portugal,634,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Inverse matrix ,Solve matrix equation,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Commuting matrices,Determinant computation" +1179,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +974,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1353,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1533,Portugal,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +625,Russian Federation,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1484,Portugal,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +700,Portugal,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1147,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +643,Italy,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1288,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1196,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1215,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +967,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +91,Lithuania,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +509,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1495,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1140,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +52,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +1077,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1023,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +972,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1352,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +91,Lithuania,117,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +380,Portugal,864,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +693,Portugal,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1163,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1565,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1185,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1312,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +634,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +855,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1179,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +37,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +509,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +985,Romania,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +722,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1044,Lithuania,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +720,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1324,Italy,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,513,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +1351,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +340,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +378,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +128,Lithuania,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +457,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +26,Portugal,546,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonal projection,Distance,Cross product,Orthogonality" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1288,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1542,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1146,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +81,Lithuania,701,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +36,Portugal,853,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +81,Lithuania,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +311,Ireland,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +552,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +26,Portugal,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1331,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +666,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +1080,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +643,Italy,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1093,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,820,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +918,Portugal,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +71,Portugal,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +374,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,771,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1517,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1179,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1557,Portugal,919,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +90,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1219,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +636,Italy,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1246,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +1244,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +776,Portugal,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1505,Portugal,298,1,Basic,Probability,Probability,"Event,Axioms of probability" +71,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +741,Ireland,478,0,Basic,Graph Theory,Graph Theory,Adjacency matrix +199,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1143,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +80,Lithuania,486,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +689,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +200,Lithuania,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +91,Lithuania,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +591,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1331,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +218,Lithuania,667,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +626,Italy,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1071,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +658,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +26,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +71,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +633,Italy,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,394,0,Basic,Integration,Integration Techniques,Substitution +189,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1543,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1308,Italy,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1201,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +422,Portugal,128,1,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1077,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +876,Italy,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1252,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +509,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +72,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1150,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1543,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1291,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +175,Italy,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +200,Lithuania,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +181,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +967,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +176,Italy,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +956,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1299,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +218,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +638,Italy,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1146,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1109,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1565,Portugal,315,0,Advanced,Integration,Integration Techniques,Integration by parts +979,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,356,1,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1322,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +598,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +411,Portugal,849,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +176,Italy,1121,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +955,Lithuania,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1561,Portugal,718,0,Advanced,Integration,Integration Techniques,Substitution +155,Spain,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1291,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +168,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +510,Portugal,141,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1543,Portugal,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +986,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +751,Lithuania,567,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +955,Lithuania,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1244,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +407,Portugal,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1241,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +465,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +80,Lithuania,917,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,270,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +26,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +28,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +510,Portugal,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +406,Portugal,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +899,Lithuania,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +351,Portugal,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +975,Portugal,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +956,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1273,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +955,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +198,Italy,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1487,Portugal,1197,1,Basic,Integration,Definite Integrals,"Area of a planar region ,Fundamental theorem of Calculus" +633,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +189,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +1564,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1166,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1077,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +776,Portugal,552,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +397,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +956,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +37,Portugal,879,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +128,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +658,Ireland,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1334,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1056,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +840,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +967,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1477,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +898,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1281,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,686,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +81,Lithuania,470,0,Advanced,Graph Theory,Graph Theory,"Simple graph,Adjacency matrix" +204,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +890,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,736,0,Basic,Optimization,Linear Optimization,"Linear programming,Excel solver add-in,Simplex method" +1076,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +176,Italy,613,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +956,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1093,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +988,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1192,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +582,Lithuania,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +168,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +85,Lithuania,374,0,Basic,Graph Theory,Graph Theory,"Tree,Spanning tree" +549,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +422,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +852,Italy,88,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1182,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1026,Lithuania,147,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +1557,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +890,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +509,Portugal,523,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +684,Ireland,211,1,Basic,Statistics,Statistics,Linear regression +189,Portugal,371,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +1079,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +311,Ireland,130,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1256,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +776,Portugal,702,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1087,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +175,Italy,1111,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +898,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1179,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +552,Portugal,683,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1289,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,828,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,698,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +1542,Portugal,1100,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +72,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1543,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +470,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +986,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +61,Portugal,889,0,Basic,Differential Equations,Differential Equations,Linear differential equation +378,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1093,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +411,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +71,Portugal,701,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product,Distance" +1269,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,221,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1244,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +81,Lithuania,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +81,Lithuania,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +91,Lithuania,520,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1289,Italy,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +955,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +168,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1273,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +955,Lithuania,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +483,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +175,Italy,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +407,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +972,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +26,Portugal,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +426,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +582,Lithuania,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +855,Italy,231,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +625,Russian Federation,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +855,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +470,Portugal,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +380,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +163,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1034,Lithuania,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +691,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +639,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1564,Portugal,921,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +351,Portugal,147,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +359,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1565,Portugal,678,1,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +852,Italy,219,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +80,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +26,Portugal,765,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules" +1557,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1269,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +176,Italy,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1150,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +26,Portugal,165,1,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1428,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +663,Portugal,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +719,Portugal,186,0,Basic,Statistics,Statistics,"Population,Sample" +181,Italy,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +657,Ireland,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1363,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1340,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +524,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1291,Italy,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1543,Portugal,1120,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +966,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +969,Romania,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,127,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +175,Italy,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +955,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +468,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +643,Italy,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1080,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +376,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +913,Ireland,492,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +717,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1284,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +37,Portugal,262,0,Basic,Differentiation,Partial Differentiation,First order +549,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1334,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +551,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +91,Lithuania,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1338,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +876,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +697,Portugal,77,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +955,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1543,Portugal,1104,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1505,Portugal,737,0,Basic,Optimization,Linear Optimization,Linear programming +91,Lithuania,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +890,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +509,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1179,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +414,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +374,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,338,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +483,Portugal,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1252,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +91,Lithuania,502,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1256,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +64,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +955,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +509,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +378,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1292,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +851,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +651,Ireland,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1028,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +663,Portugal,78,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +426,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +973,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +393,Lithuania,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +890,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1109,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +958,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1275,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1516,Portugal,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +168,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +240,Lithuania,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +71,Portugal,1061,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1252,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,1100,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +198,Italy,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +852,Italy,243,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1272,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +64,Portugal,551,0,Basic,Analytic Geometry,Analytic Geometry,Orthogonality +175,Italy,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +552,Portugal,928,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +80,Lithuania,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +851,Italy,1247,0,Basic,Optimization,Nonlinear Optimization,"Unconstrained optimization,Second derivative test,Optimal solution" +853,Italy,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,225,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +41,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +898,Portugal,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +455,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +855,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +651,Ireland,186,0,Basic,Statistics,Statistics,"Population,Sample" +969,Romania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1284,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1070,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1324,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,679,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +421,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +37,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +197,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1518,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1076,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1229,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +955,Lithuania,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1487,Portugal,272,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +168,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +979,Portugal,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1196,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +393,Lithuania,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +168,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1252,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1146,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +37,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +85,Lithuania,1157,0,Basic,Set Theory,Set Theory,Subset +1179,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1505,Portugal,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +374,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +359,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +41,Portugal,507,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +176,Italy,108,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +41,Portugal,561,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +72,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +667,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +853,Italy,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +1292,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +473,Portugal,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +131,Lithuania,586,1,Advanced,Graph Theory,Graph Theory,Adjacency matrix +448,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1292,Italy,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +81,Lithuania,556,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +41,Portugal,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +378,Portugal,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1148,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +155,Spain,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1135,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1076,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +340,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +340,Portugal,320,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +625,Russian Federation,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +753,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1146,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +626,Italy,274,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1557,Portugal,694,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1197,Slovenia,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1518,Portugal,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1273,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1093,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +311,Ireland,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +69,Portugal,556,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line,Collinearity" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +36,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1150,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +181,Italy,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +394,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1331,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1150,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1271,Portugal,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +90,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +28,Portugal,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1140,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1010,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum" +918,Portugal,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1219,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +81,Lithuania,938,1,Advanced,Graph Theory,Graph Theory,Chromatic number +26,Portugal,697,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1517,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1271,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1146,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +599,Italy,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +853,Italy,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +72,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +979,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1525,Portugal,740,0,Advanced,Optimization,Linear Optimization,Linear programming +80,Lithuania,282,0,Basic,Optimization,Linear Optimization,Linear programming +1543,Portugal,1105,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1087,Portugal,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1351,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +855,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +174,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +973,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +37,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +473,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1253,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +72,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +965,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1351,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1271,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1197,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +394,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +403,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +642,Italy,211,1,Basic,Statistics,Statistics,Linear regression +1077,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +966,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +899,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +71,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1023,Lithuania,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,821,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Second derivative test,Stationary point,Unconstrained optimization,Saddle point" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1256,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +426,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +693,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +851,Italy,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +972,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +380,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1518,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1219,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +666,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +972,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +600,Italy,138,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +91,Lithuania,500,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1375,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1080,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +57,Portugal,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +851,Italy,1007,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Stationary point,Maximum,Saddle point" +1351,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +378,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1517,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +151,Portugal,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1339,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1252,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,568,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +969,Romania,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1252,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1044,Lithuania,880,0,Advanced,Differentiation,Partial Differentiation,"First order,Directional derivative,Gradient" +1322,Italy,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +955,Lithuania,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1034,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +956,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1163,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +448,Portugal,310,1,Advanced,Probability,Probability,"Direct inspection,Simple integration" +91,Lithuania,521,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +349,Portugal,271,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +509,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +483,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +406,Portugal,215,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +884,Italy,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1274,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +751,Lithuania,697,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Cross product" +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +955,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,698,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Distance" +663,Portugal,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1350,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +625,Russian Federation,664,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +166,Portugal,646,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +1022,Lithuania,959,0,Basic,Optimization,Linear Optimization,Linear programming +876,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +409,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +72,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +468,Portugal,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +955,Lithuania,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +853,Italy,215,0,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Inconsistent system,Solution of linear system" +692,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +37,Portugal,192,0,Basic,Integration,Integration Techniques,"Trigonometric substitution,Simple integration" +1557,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +351,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +422,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +189,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +524,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +465,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +552,Portugal,868,1,Basic,Differential Equations,Differential Equations,Separable variables equation +1109,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +468,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +890,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,516,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1179,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1109,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +955,Lithuania,1101,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +510,Portugal,608,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +359,Portugal,501,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1557,Portugal,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1230,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +552,Portugal,87,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +669,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1276,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +359,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +985,Romania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +967,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1350,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1341,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +239,Lithuania,273,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +91,Lithuania,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +1220,Slovenia,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1543,Portugal,1106,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +168,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +660,Portugal,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +741,Ireland,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +972,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1023,Lithuania,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1194,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +311,Ireland,644,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +633,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1289,Italy,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +380,Portugal,871,0,Basic,Differential Equations,Differential Equations,"Cauchy problem,Exact differential form" +1511,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +525,Portugal,518,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +37,Portugal,153,1,Advanced,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +411,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +890,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1284,Italy,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1039,Lithuania,258,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +955,Lithuania,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +311,Ireland,653,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +851,Italy,858,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Minimum,Maximum,Stationary point,Saddle point" +851,Italy,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +672,Portugal,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +426,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +155,Spain,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +510,Portugal,850,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1351,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +556,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1143,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1135,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1271,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +625,Russian Federation,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1521,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Quadratic equations,Simplify expressions,Linear equations" +884,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1289,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +986,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +168,Portugal,678,0,Basic,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1080,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1082,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1517,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +535,Portugal,354,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +468,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +374,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +466,Portugal,333,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +80,Lithuania,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1023,Lithuania,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1487,Portugal,940,0,Basic,Graph Theory,Graph Theory,"Spanning tree,Shortest path,Path,Tree" +426,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1087,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,643,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +1122,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1324,Italy,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,916,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +576,Ireland,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1339,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1140,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +406,Portugal,787,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +625,Russian Federation,517,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +91,Lithuania,516,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +80,Lithuania,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +174,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1411,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +128,Lithuania,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1525,Portugal,769,0,Advanced,Probability,Probability,"Bayes' Theorem,Conditional probability" +1256,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +91,Lithuania,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1038,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +69,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +509,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +576,Ireland,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +41,Portugal,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +61,Portugal,888,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1402,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1284,Italy,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1220,Slovenia,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1049,Lithuania,657,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +241,Lithuania,603,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +397,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1481,Portugal,714,1,Basic,Integration,Integration Techniques,"Simple integration,Direct inspection" +1271,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,597,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +351,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1277,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +634,Italy,273,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +633,Italy,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +91,Lithuania,495,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1229,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1428,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +967,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +525,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +406,Portugal,497,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1538,Portugal,1097,0,Basic,Numerical Methods,Numerical Methods,"Lagrange's interpolation,Polynomial interpolation" +955,Lithuania,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +973,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1505,Portugal,1003,0,Basic,Optimization,Linear Optimization,Linear programming +351,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1269,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +344,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1271,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +394,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +854,Italy,89,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +240,Lithuania,922,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1487,Portugal,332,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +168,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +712,Portugal,185,0,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1150,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +394,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +674,Ireland,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +90,Lithuania,951,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +690,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +966,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +168,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1256,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1543,Portugal,1112,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +753,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +406,Portugal,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1487,1,Basic,Complex Numbers,Complex Numbers,"Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers,Imaginary part,Conjugate number" +380,Portugal,865,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +80,Lithuania,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +57,Portugal,190,1,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1194,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +972,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1219,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +851,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +28,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +669,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +720,Portugal,193,1,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +626,Italy,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +1056,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1081,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1157,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +91,Lithuania,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +198,Italy,364,0,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +451,Portugal,318,1,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1271,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +411,Portugal,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1081,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1023,Lithuania,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,862,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +1487,Portugal,1060,0,Basic,Complex Numbers,Complex Numbers,Complex plane +898,Portugal,864,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Optimal solution,Minimum,Maximum" +28,Portugal,180,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +624,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1091,Lithuania,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1243,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1043,Lithuania,873,0,Basic,Differentiation,Partial Differentiation,First order +465,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +470,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +626,Italy,921,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1247,Portugal,349,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +81,Lithuania,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +965,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1334,Portugal,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1220,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +176,Italy,104,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +457,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1158,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,500,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Triangles,Classification of geometrical figures" +1286,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +168,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +41,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1305,Italy,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +483,Portugal,340,1,Advanced,Complex Numbers,Complex Numbers,"Operations with complex numbers,Conjugate number" +155,Spain,157,1,Basic,Differential Equations,Differential Equations,"Homogeneous equation,Cauchy problem" +634,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1271,Portugal,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +28,Portugal,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +511,Portugal,130,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +448,Portugal,343,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1557,Portugal,889,1,Basic,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1158,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1182,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +524,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +37,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +986,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1525,Portugal,860,0,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +972,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1185,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +510,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +988,Portugal,916,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Square linear system,Consistent system" +407,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1487,Portugal,729,1,Basic,Differentiation,Derivatives,Quotient rule +972,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +974,Portugal,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1292,Italy,622,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +913,Ireland,1131,0,Basic,Set Theory,Set Theory,Subset +898,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,467,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Polygons,Classification of geometrical figures,Geometrical figures" +41,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1338,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1542,Portugal,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1026,Lithuania,729,1,Basic,Differentiation,Derivatives,Quotient rule +1543,Portugal,1109,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1081,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +219,Lithuania,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +625,Russian Federation,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1140,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1195,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1229,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +403,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1081,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +160,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +81,Lithuania,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +380,Portugal,890,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1288,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +344,Portugal,720,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules,Logarithmic rule,Product rule" +1517,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1077,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1351,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1525,Portugal,763,0,Advanced,Probability,Probability,Conditional probability +697,Portugal,178,1,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +913,Ireland,1118,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1253,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1557,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +1289,Italy,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +90,Lithuania,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +913,Ireland,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +168,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1486,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +972,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +71,Portugal,653,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix rank" +26,Portugal,549,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +1023,Lithuania,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +511,Portugal,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +712,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1517,Portugal,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +524,Portugal,668,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +64,Portugal,309,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +375,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +311,Ireland,645,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,507,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +972,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1333,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +468,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +71,Portugal,786,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1241,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1185,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +177,Italy,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +26,Portugal,567,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a line,Orthogonality,Collinearity" +741,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +638,Italy,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1093,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +37,Portugal,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +175,Italy,1096,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +852,Italy,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1093,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +634,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1289,Italy,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +414,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +852,Italy,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +876,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1044,Lithuania,1002,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +898,Portugal,1053,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1077,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1173,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1319,Italy,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +204,Lithuania,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +625,Russian Federation,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +625,Russian Federation,799,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +411,Portugal,624,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +468,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1091,Lithuania,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +750,Portugal,83,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +177,Italy,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +1087,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +177,Italy,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1324,Italy,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1244,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +576,Ireland,572,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Scalar product,Cross product,Distance,Collinearity,Orthogonality" +972,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +674,Ireland,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +510,Portugal,652,0,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +406,Portugal,123,0,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +80,Lithuania,752,1,Basic,Differentiation,Derivatives,"Power rule,Quotient rule,Product rule,Trigonometric rules" +851,Italy,155,1,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +525,Portugal,112,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +955,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +973,Portugal,135,1,Basic,Linear Algebra,Matrices and Determinants,"Symmetric matrix,Determinant computation,Determinant properties" +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +525,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1292,Italy,849,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +876,Italy,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,1063,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +349,Portugal,275,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +26,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +689,Portugal,860,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +72,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +36,Portugal,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1146,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1185,Portugal,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1182,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +26,Portugal,878,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +71,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1147,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +455,Portugal,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +473,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1147,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1197,Slovenia,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1023,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1185,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +913,Ireland,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +91,Lithuania,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +239,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +1076,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1079,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1147,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +958,Lithuania,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +553,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +582,Lithuania,562,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +851,Italy,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +854,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1288,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +151,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +376,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +986,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +128,Lithuania,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +524,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1543,Portugal,1115,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,188,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +625,Russian Federation,792,1,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1543,Portugal,1110,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1093,Portugal,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +359,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +81,Lithuania,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1428,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +470,Portugal,351,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,1090,1,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +918,Portugal,399,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +174,Portugal,366,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1122,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1044,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1023,Lithuania,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +26,Portugal,573,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Locus,Scalar product,Cross product,Collinearity" +852,Italy,949,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1427,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,1094,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +351,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +966,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +1505,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +175,Italy,1119,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +913,Ireland,1102,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,1115,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1146,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +524,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +174,Portugal,336,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +1219,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +884,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +394,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1195,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1252,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +591,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +776,Portugal,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +966,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +151,Portugal,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +419,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1076,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +510,Portugal,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +72,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +380,Portugal,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +851,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +582,Lithuania,575,0,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1516,Portugal,891,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1032,Lithuania,877,1,Basic,Differentiation,Partial Differentiation,Second order +168,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +855,Italy,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1334,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +979,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +168,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +576,Ireland,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +165,Portugal,391,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +955,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +483,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +913,Ireland,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1292,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +465,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +956,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +344,Portugal,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1173,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +163,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1150,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1044,Lithuania,874,0,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule" +1147,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +175,Italy,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +625,Russian Federation,121,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1182,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1288,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1095,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +851,Italy,963,0,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +240,Lithuania,919,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1288,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1080,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +510,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +956,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +898,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,523,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Polygons" +509,Portugal,244,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +80,Lithuania,917,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +175,Italy,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +81,Lithuania,582,1,Advanced,Graph Theory,Graph Theory,"Path,Cycle and circuit" +509,Portugal,636,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +979,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +972,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +409,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +955,Lithuania,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +351,Portugal,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +340,Portugal,300,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1034,Lithuania,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +969,Romania,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +852,Italy,242,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +633,Italy,132,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1087,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +548,Portugal,300,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +689,Portugal,964,1,Basic,Optimization,Nonlinear Optimization,"Simplex method,Duality theory" +1543,Portugal,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +199,Lithuania,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1282,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1518,Portugal,637,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +85,Lithuania,1151,0,Basic,Set Theory,Set Theory,Subset +1288,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +885,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +175,Italy,1497,1,Basic,Complex Numbers,Complex Numbers,"Imaginary part,Conjugate number,Modulus of a complex number,Algebraic form,Real part,Equations involving complex numbers" +511,Portugal,112,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +642,Italy,189,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +985,Romania,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +956,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +36,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +913,Ireland,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1044,Lithuania,998,1,Basic,Differentiation,Partial Differentiation,First order +982,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1299,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1341,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,462,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +151,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +657,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1230,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +899,Lithuania,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1319,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1047,Lithuania,259,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +451,Portugal,344,1,Advanced,Probability,Probability,"Event,Independence,Addition, multiplication, and total probability rules" +986,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +28,Portugal,557,1,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +359,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +524,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +511,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +966,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1516,Portugal,892,1,Basic,Differential Equations,Differential Equations,Linear differential equation +91,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,911,1,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +576,Ireland,568,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,98,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +41,Portugal,79,1,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +175,Italy,1549,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Linear equations,Quadratic equations" +1077,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +411,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +176,Italy,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +625,Russian Federation,614,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1339,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +853,Italy,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +470,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +972,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +1081,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1046,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +91,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +684,Ireland,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +426,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +421,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +378,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +198,Italy,367,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +91,Lithuania,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +37,Portugal,1032,1,Basic,Differentiation,Partial Differentiation,"Power rule,Trigonometric rules,Logarithmic rule,First order" +582,Lithuania,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1543,Portugal,1093,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1157,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1248,Italy,234,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1375,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +465,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1256,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +219,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,144,0,Basic,Differentiation,Derivatives,Trigonometric rules +71,Portugal,136,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1087,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1518,Portugal,141,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +189,Portugal,665,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1116,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1411,Portugal,781,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1075,Portugal,148,0,Basic,Differentiation,Partial Differentiation,First order +1244,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1291,Italy,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +658,Ireland,79,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +840,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,97,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Logarithmic rule,Power rule" +37,Portugal,1230,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +884,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +72,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1037,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +72,Portugal,385,0,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1024,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +457,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1220,Slovenia,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +899,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +311,Ireland,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +378,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1197,Slovenia,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +955,Lithuania,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,416,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +972,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +1305,Italy,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1081,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +988,Portugal,216,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1143,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1505,Portugal,955,0,Basic,Optimization,Linear Optimization,Linear programming +175,Italy,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +64,Portugal,253,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +1517,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +378,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,506,1,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +128,Lithuania,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +71,Portugal,852,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1182,Portugal,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +155,Spain,165,0,Basic,Differential Equations,Differential Equations,"Linear differential equation,Homogeneous equation" +551,Lithuania,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +483,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +41,Portugal,562,1,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +913,Ireland,1108,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +667,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1135,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1290,Italy,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +599,Italy,628,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue,Diagonalization" +967,Portugal,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1158,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +840,Italy,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +625,Russian Federation,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +1173,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1130,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +876,Italy,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1322,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1339,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +751,Lithuania,999,0,Basic,Differentiation,Partial Differentiation,First order +1171,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +397,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1290,Italy,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +411,Portugal,856,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1032,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +80,Lithuania,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +876,Italy,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +473,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +91,Lithuania,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +69,Portugal,565,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonal projection,Orthogonality,Collinearity,Locus" +1533,Portugal,564,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +988,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +509,Portugal,635,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +175,Italy,1252,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +956,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1564,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +840,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1275,Lithuania,440,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +64,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +37,Portugal,886,1,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,394,1,Basic,Integration,Integration Techniques,Substitution +26,Portugal,217,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +204,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +80,Lithuania,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +340,Portugal,304,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +426,Portugal,351,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Imaginary part,Complex plane ,Operations with complex numbers,Real part,Principal argument,De Moivre formulas" +1082,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1179,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,88,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1269,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +851,Italy,911,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1288,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1322,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1300,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1077,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1279,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +548,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +510,Portugal,368,0,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1195,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +669,Portugal,81,0,Basic,Statistics,Statistics,"Mean,Median,Mode" +1269,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +28,Portugal,331,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +913,Ireland,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +641,Italy,186,0,Basic,Statistics,Statistics,"Population,Sample" +1352,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +351,Portugal,788,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +411,Portugal,616,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1229,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1517,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +956,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1533,Portugal,557,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance" +1185,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +468,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +966,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +876,Italy,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +769,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +250,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1248,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1194,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +80,Lithuania,757,0,Basic,Integration,Double Integration,"Iterate integrals,Cartesian coordinates,y-simple region (type I)" +1076,Portugal,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +979,Portugal,1120,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1271,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1487,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +913,Ireland,211,0,Basic,Statistics,Statistics,Linear regression +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1517,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +64,Portugal,406,0,Basic,Integration,Integration Techniques,Direct inspection +90,Lithuania,641,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +979,Portugal,1107,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1195,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1379,Portugal,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +1220,Slovenia,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1241,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +890,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +913,Ireland,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +422,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +697,Portugal,201,0,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +1220,Slovenia,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +351,Portugal,918,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1509,Portugal,83,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics","Domain,Image of a function" +41,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1243,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1158,Portugal,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +483,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1276,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1517,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,517,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +525,Portugal,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,519,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +884,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +349,Portugal,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +625,Russian Federation,113,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1045,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +1023,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1299,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +967,Portugal,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1023,Lithuania,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +637,Italy,182,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +1286,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +168,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +166,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +509,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +851,Italy,861,1,Basic,Optimization,Nonlinear Optimization,"Saddle point,Optimal solution,Minimum,Maximum,Stationary point" +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1087,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1291,Italy,781,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +168,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +473,Portugal,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +61,Portugal,837,0,Basic,Differential Equations,Differential Equations,Linear differential equation +1332,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +61,Portugal,870,1,Basic,Differential Equations,Differential Equations,Homogeneous equation +1135,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1023,Lithuania,455,0,Basic,Linear Algebra,Vector Spaces,Basis +403,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1479,Portugal,955,1,Basic,Optimization,Linear Optimization,Linear programming +168,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,794,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +175,Italy,1117,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1219,Portugal,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1505,Portugal,189,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +168,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +775,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +634,Italy,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1230,Portugal,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1351,Portugal,126,1,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +775,Portugal,308,1,Basic,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +1375,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1023,Lithuania,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +898,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1275,Lithuania,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +667,Portugal,202,1,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +351,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +468,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1352,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +1352,Portugal,290,1,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1022,Lithuania,738,0,Basic,Optimization,Linear Optimization,"Graphical method,Linear programming" +1023,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +359,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +91,Lithuania,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +91,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +380,Portugal,169,0,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1334,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1114,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +852,Italy,246,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1220,Slovenia,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +378,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +411,Portugal,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +913,Ireland,492,1,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Axioms of probability,Event" +1308,Italy,782,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +359,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1308,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +71,Portugal,771,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +394,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +899,Lithuania,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1143,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1286,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +91,Lithuania,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +851,Italy,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +351,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +380,Portugal,868,0,Basic,Differential Equations,Differential Equations,Separable variables equation +36,Portugal,607,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1279,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +1171,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1525,Portugal,295,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1163,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +966,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +81,Lithuania,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +81,Lithuania,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +462,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1379,Portugal,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +972,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +741,Ireland,935,0,Basic,Graph Theory,Graph Theory,Chromatic number +1288,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +351,Portugal,612,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +986,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +26,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +168,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +955,Lithuania,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1087,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +591,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +876,Italy,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1219,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +420,Portugal,327,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1044,Lithuania,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +26,Portugal,1157,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1195,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1543,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +827,Italy,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +535,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +852,Italy,239,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +851,Italy,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1274,Portugal,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1350,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1286,Italy,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +80,Lithuania,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1071,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1290,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1564,Portugal,680,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1289,Italy,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +884,Italy,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1077,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +473,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1093,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +239,Lithuania,255,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Logarithmic rule" +958,Lithuania,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +422,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1219,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +884,Italy,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +639,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +168,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +80,Lithuania,788,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Exponential function" +451,Portugal,307,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1557,Portugal,926,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +625,Russian Federation,495,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +61,Portugal,831,0,Basic,Differential Equations,Differential Equations,Linear differential equation +37,Portugal,1084,1,Advanced,Differentiation,Partial Differentiation,"First order,Second order" +625,Russian Federation,522,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Triangles" +1147,Portugal,455,1,Basic,Linear Algebra,Vector Spaces,Basis +376,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +986,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +91,Lithuania,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1324,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1143,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +955,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +988,Portugal,950,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1197,Slovenia,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +890,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1352,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1157,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +177,Italy,730,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +91,Lithuania,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1080,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +851,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1256,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +1220,Slovenia,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1312,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +913,Ireland,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,1076,0,Basic,Differentiation,Partial Differentiation,"Logarithmic rule,Power rule,Trigonometric rules" +1241,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1220,Slovenia,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1044,Lithuania,875,1,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1081,Portugal,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +411,Portugal,335,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +376,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +406,Portugal,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +81,Lithuania,481,1,Advanced,Graph Theory,Graph Theory,"Path,Simple graph,Cycle and circuit" +375,Portugal,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1252,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +168,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +175,Italy,1514,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1220,Slovenia,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1087,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +128,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1220,Slovenia,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +986,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +422,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +591,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +175,Italy,1093,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +376,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +913,Ireland,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +884,Italy,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +374,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +552,Portugal,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +1288,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1049,Lithuania,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +1543,Portugal,1086,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1288,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +414,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +394,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +913,Ireland,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +64,Portugal,120,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +131,Lithuania,117,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1269,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +751,Lithuania,703,1,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +1288,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +511,Portugal,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1220,Slovenia,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +81,Lithuania,937,0,Advanced,Graph Theory,Graph Theory,Chromatic number +753,Italy,1000,0,Basic,Differentiation,Partial Differentiation,First order +202,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +376,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +651,Ireland,187,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +414,Portugal,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +855,Italy,218,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +965,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +36,Portugal,1133,1,Basic,Set Theory,Set Theory,Subset +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +913,Ireland,1251,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +899,Lithuania,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +974,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1087,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1299,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1517,Portugal,418,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +974,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +155,Spain,148,1,Basic,Differentiation,Partial Differentiation,First order +1173,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1252,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +69,Portugal,353,1,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1135,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +1023,Lithuania,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +955,Lithuania,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +524,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1220,Slovenia,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +525,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1322,Italy,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1109,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +90,Lithuania,355,0,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +1087,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,595,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +988,Portugal,233,1,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +1427,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +700,Portugal,201,1,Basic,Statistics,Statistics,"Chi square distribution,Student distribution,Quantile,Hypothesis testing" +69,Portugal,548,1,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Distance,Locus,Cross product,Orthogonality,Scalar product" +349,Portugal,82,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Exponential rule,Trigonometric rules,Product rule" +387,Portugal,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1487,Portugal,1164,1,Basic,Integration,Definite Integrals,"Direct integrals ,Fundamental theorem of Calculus" +411,Portugal,414,0,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Integration by parts,Rational functions" +1332,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +394,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +198,Italy,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +599,Italy,594,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Spectrum,Characteristic polynomial,Diagonalization,Eigenvalue" +91,Lithuania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1082,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,321,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +57,Portugal,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +71,Portugal,123,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +71,Portugal,753,1,Basic,Differentiation,Derivatives,Exponential rule +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1517,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1308,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +986,Portugal,418,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1505,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +351,Portugal,100,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule,Sum rule,Power rule" +1109,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +552,Portugal,575,1,Advanced,Analytic Geometry,Analytic Geometry,"Orthogonality,Orthogonal projection,Cross product,Scalar product" +899,Lithuania,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1075,Portugal,908,1,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +956,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1077,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1044,Lithuania,263,1,Basic,Differentiation,Partial Differentiation,First order +1557,Portugal,693,1,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1220,Slovenia,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1483,Portugal,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1305,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1081,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +468,Portugal,333,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +751,Lithuania,734,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Exponential rule" +1039,Lithuania,264,0,Basic,Differentiation,Partial Differentiation,Chain rule +973,Portugal,630,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +776,Portugal,564,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Scalar product,Cross product,Orthogonality" +1081,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +851,Italy,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +359,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +28,Portugal,563,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +483,Portugal,368,1,Basic,Fundamental Mathematics,Elementary Geometry,"Platonic solids,Regular polygon" +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +90,Lithuania,126,0,Advanced,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Solve matrix equation" +641,Italy,187,1,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +854,Italy,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1039,Lithuania,908,0,Basic,Differentiation,Partial Differentiation,"Gradient,Directional derivative" +840,Italy,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +511,Portugal,221,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +37,Portugal,399,1,Basic,Integration,Integration Techniques,"Substitution,Direct inspection,Simple integration" +840,Italy,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +407,Portugal,329,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +661,Portugal,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1481,Portugal,681,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +387,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,357,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +1076,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +80,Lithuania,229,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +414,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +551,Lithuania,342,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +71,Portugal,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1517,Portugal,451,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +175,Italy,1473,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +633,Italy,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +175,Italy,327,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +80,Lithuania,756,1,Basic,Integration,Double Integration,"Cartesian coordinates,x-simple region (type II),Changing order of integration" +1146,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1334,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,836,1,Basic,Differential Equations,Differential Equations,Linear differential equation +483,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +955,Lithuania,1109,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +1135,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +751,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,136,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +988,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1049,Lithuania,148,0,Basic,Differentiation,Partial Differentiation,First order +1219,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +898,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +898,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1341,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +376,Portugal,1056,1,Advanced,Complex Numbers,Complex Numbers,Complex plane +956,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +91,Lithuania,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1095,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1256,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1322,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1286,Italy,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1486,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +966,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +199,Lithuania,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +165,Portugal,875,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1076,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +626,Italy,753,1,Basic,Differentiation,Derivatives,Exponential rule +969,Romania,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +473,Portugal,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +91,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +840,Italy,778,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1244,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +405,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +697,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1274,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1533,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +174,Portugal,501,0,Basic,Fundamental Mathematics,Elementary Geometry,Geometric solids +203,Lithuania,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1079,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +411,Portugal,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1256,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1292,Italy,629,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +394,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1564,Portugal,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +81,Lithuania,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +1087,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1179,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +634,Italy,274,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +1081,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1273,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +972,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +1220,Slovenia,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1093,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +876,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1220,Slovenia,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +406,Portugal,223,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +979,Portugal,1096,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +899,Lithuania,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1229,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +966,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +72,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +168,Portugal,777,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +448,Portugal,322,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +198,Italy,613,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1163,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +451,Portugal,343,1,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +1091,Lithuania,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +394,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +414,Portugal,331,1,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,De Moivre formulas,Operations with complex numbers,Imaginary part" +1050,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +71,Portugal,271,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1194,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1028,Lithuania,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1150,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +598,Italy,222,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1271,Portugal,779,0,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +600,Italy,131,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,566,0,Basic,Analytic Geometry,Analytic Geometry,"Distance,Orthogonality,Locus" +689,Portugal,867,0,Basic,Optimization,Nonlinear Optimization,"Stationary point,Maximum,Saddle point,Second derivative test,Optimal solution,Minimum" +1411,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +81,Lithuania,381,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +853,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1192,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1292,Italy,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +510,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1076,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +633,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1322,Italy,455,0,Basic,Linear Algebra,Vector Spaces,Basis +359,Portugal,506,0,Basic,Fundamental Mathematics,Elementary Geometry,Symmetry +552,Portugal,574,0,Advanced,Analytic Geometry,Analytic Geometry,"Collinearity,Orthogonality,Distance,Scalar product" +511,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +983,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +457,Portugal,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1081,Portugal,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +918,Portugal,397,0,Basic,Integration,Integration Techniques,Substitution +955,Lithuania,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1171,Portugal,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +375,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +80,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +852,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +155,Spain,153,1,Basic,Differentiation,Partial Differentiation,"Stationary point,Minimum,Optimal solution,Lagrange multipliers" +1289,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,883,0,Basic,Differentiation,Partial Differentiation,"First order,Directional derivative" +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +840,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +37,Portugal,1071,1,Basic,Differentiation,Partial Differentiation,First order +1023,Lithuania,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +913,Ireland,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +37,Portugal,851,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +509,Portugal,376,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +634,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +349,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +741,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +509,Portugal,602,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +840,Italy,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +853,Italy,228,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1286,Italy,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +411,Portugal,401,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +468,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +967,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +383,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +151,Portugal,102,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +168,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +448,Portugal,304,1,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +175,Italy,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1220,Slovenia,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1292,Italy,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +414,Portugal,220,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +913,Ireland,294,0,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +1517,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +426,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +840,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1087,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1158,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +510,Portugal,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +359,Portugal,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/val.csv b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/val.csv new file mode 100644 index 0000000000000000000000000000000000000000..3e038bf8c24a2f9b2a1d9ee36e4e68c500c7f024 --- /dev/null +++ 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Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1135,Portugal,278,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +524,Portugal,496,0,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1023,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +913,Ireland,1252,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +61,Portugal,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1220,Slovenia,379,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1288,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +664,Portugal,178,0,Basic,Statistics,Statistics,"Mean,Variance,Standard deviation,Median,Mode" +986,Portugal,777,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation 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method" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +81,Lithuania,480,1,Basic,Graph Theory,Graph Theory,Planar graph +1543,Portugal,1114,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +457,Portugal,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +80,Lithuania,668,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +415,Portugal,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1274,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +509,Portugal,605,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1135,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +625,Russian Federation,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1229,Portugal,290,0,Advanced,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1284,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +151,Portugal,113,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +633,Italy,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +582,Lithuania,700,0,Basic,Analytic Geometry,Analytic Geometry,"Collinearity,Locus,Orthogonality,Orthogonal projection,Scalar product,Distance" +37,Portugal,773,0,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +91,Lithuania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +90,Lithuania,139,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication,Determinant computation,Determinant properties" +1543,Portugal,1113,0,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +511,Portugal,464,0,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +1022,Lithuania,279,0,Basic,Optimization,Linear Optimization,Linear programming +1334,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1046,Lithuania,877,0,Basic,Differentiation,Partial Differentiation,Second order +1146,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +200,Lithuania,394,1,Basic,Integration,Integration Techniques,Substitution +483,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,508,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1173,Portugal,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1147,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +81,Lithuania,471,1,Advanced,Graph Theory,Graph Theory,Simple graph +175,Italy,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1283,Italy,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +470,Portugal,347,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +344,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +524,Portugal,231,1,Basic,Linear Algebra,Linear Systems,"Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system,Solution of linear system" +91,Lithuania,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +884,Italy,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1194,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +91,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +639,Italy,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +1272,Portugal,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1341,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1292,Italy,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +71,Portugal,342,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Modulus of a complex number,Algebraic form" +155,Spain,143,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +1147,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +175,Italy,1087,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1284,Italy,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1286,Italy,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +375,Portugal,358,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +956,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +198,Italy,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +411,Portugal,413,0,Basic,Integration,Integration Techniques,Direct inspection +41,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +898,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +181,Italy,925,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1182,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +525,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1253,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +407,Portugal,729,0,Basic,Differentiation,Derivatives,Quotient rule +36,Portugal,143,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Logarithmic rule" +464,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +663,Portugal,180,1,Basic,Statistics,Statistics,"Mean,Median,Mode" +1023,Lithuania,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +955,Lithuania,1090,0,Basic,Numerical Methods,Numerical Methods,"Nonlinear equation,Newton’s method" +1350,Portugal,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1286,Italy,426,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1428,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +582,Lithuania,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +898,Portugal,432,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +639,Italy,77,0,Basic,Statistics,Statistics,"Stem and Leaf diagram,Relative frequency,Sample,Frequency" +1093,Portugal,380,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1220,Slovenia,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +80,Lithuania,244,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1289,Italy,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace +913,Ireland,307,1,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Independence" +1284,Italy,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1109,Portugal,434,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +511,Portugal,91,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1272,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1339,Portugal,430,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +311,Ireland,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +91,Lithuania,518,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +668,Portugal,204,0,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +1340,Portugal,1112,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +634,Italy,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1271,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +634,Italy,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1351,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +311,Ireland,122,1,Basic,Linear Algebra,Matrices and Determinants,"Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication,Determinant properties,Symmetric matrix" +979,Portugal,1104,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +958,Lithuania,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1091,Lithuania,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +473,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1087,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +661,Portugal,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +753,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1082,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1334,Portugal,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +582,Lithuania,696,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Intersection,Cartesian equations of a line" +1517,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +626,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +91,Lithuania,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +473,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +984,Romania,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +483,Portugal,93,1,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +155,Spain,169,1,Basic,Differential Equations,Differential Equations,Linear differential equation +986,Portugal,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +1045,Lithuania,912,0,Basic,Differentiation,Partial Differentiation,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +890,Portugal,432,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Subspace,Linear combination,Linear dependence,Span" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +155,Spain,869,0,Basic,Differential Equations,Differential Equations,Separable variables equation +884,Italy,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,1212,1,Basic,Statistics,Statistics,"Mean,Median,Outliers,Interquartile range" +509,Portugal,241,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +625,Russian Federation,509,0,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +359,Portugal,443,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +750,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +510,Portugal,795,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +311,Ireland,647,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +1253,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +175,Italy,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +974,Portugal,1099,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +509,Portugal,246,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1487,Portugal,1170,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Direct integrals" +407,Portugal,337,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +840,Italy,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +359,Portugal,447,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +91,Lithuania,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +57,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +64,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +174,Portugal,329,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +524,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +451,Portugal,294,1,Basic,Probability,Probability,"Event,Axioms of probability,Sample space" +457,Portugal,359,1,Advanced,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +57,Portugal,177,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +80,Lithuania,371,1,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Geometrical figures" +451,Portugal,302,0,Basic,Probability,Probability,"Addition, multiplication, and total probability rules,Sample space,Event,Axioms of probability" +359,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +1333,Portugal,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +168,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +52,Portugal,129,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +890,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +204,Lithuania,102,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +215,Lithuania,834,1,Basic,Differential Equations,Differential Equations,Linear differential equation +376,Portugal,346,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1292,Italy,602,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector,Spectrum,Characteristic polynomial" +91,Lithuania,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +1056,Lithuania,885,1,Basic,Differentiation,Partial Differentiation,"Second order,First order" +1420,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +422,Portugal,641,1,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties,Hermitian matrix,Symmetric matrix,Matrix multiplication,Linearly independent rows,Matrix operations" +626,Italy,923,1,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +57,Portugal,184,1,Basic,Statistics,Statistics,"Frequency,Sample,Stem and Leaf diagram,Variance,Standard deviation,Relative frequency" +1311,Italy,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +80,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +71,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +986,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +462,Lithuania,636,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Matrix multiplication" +851,Italy,1012,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +1220,Slovenia,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +239,Lithuania,266,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +898,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +700,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +855,Italy,223,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +28,Portugal,552,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Cross product" +91,Lithuania,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1080,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1273,Portugal,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +636,Italy,190,0,Basic,Statistics,Statistics,"Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance,Median" +1557,Portugal,673,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +219,Lithuania,789,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +397,Portugal,357,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Algebraic form,Conjugate number,Real part,Equations involving complex numbers" +69,Portugal,571,1,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1324,Italy,415,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1182,Portugal,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1147,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +776,Portugal,570,0,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +1542,Portugal,1117,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +1182,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1350,Portugal,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1322,Italy,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +376,Portugal,1060,1,Basic,Complex Numbers,Complex Numbers,Complex plane +1039,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1525,Portugal,774,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +955,Lithuania,378,1,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1252,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +524,Portugal,367,1,Basic,Fundamental Mathematics,Elementary Geometry,Geometrical figures +1220,Slovenia,455,1,Basic,Linear Algebra,Vector Spaces,Basis +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1242,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1427,Portugal,385,1,Advanced,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +955,Lithuania,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1322,Italy,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1194,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +483,Portugal,356,0,Advanced,Complex Numbers,Complex Numbers,Operations with complex numbers +1277,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +986,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +625,Russian Federation,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +1561,Portugal,411,1,Advanced,Integration,Integration Techniques,Integration by parts +1076,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +913,Ireland,961,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Homogeneous system,Consistent system" +913,Ireland,773,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +986,Portugal,306,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +955,Lithuania,1213,0,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +451,Portugal,299,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +884,Italy,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1023,Lithuania,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +468,Portugal,785,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +37,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +876,Italy,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1093,Portugal,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +509,Portugal,776,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1339,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +168,Portugal,449,1,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +1322,Italy,438,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +457,Portugal,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +840,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +52,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1479,Portugal,1004,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +1077,Portugal,449,0,Advanced,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis" +349,Portugal,272,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,630,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +525,Portugal,461,0,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1220,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +168,Portugal,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +91,Lithuania,510,1,Advanced,Fundamental Mathematics,Elementary Geometry,Symmetry +1427,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1037,Lithuania,268,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +1047,Lithuania,258,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +414,Portugal,360,1,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1565,Portugal,981,0,Basic,Real Functions of a single variable,Limits and Continuity,Limits +28,Portugal,229,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +634,Italy,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1543,Portugal,1099,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +175,Italy,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1182,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +201,Lithuania,114,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1076,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1542,Portugal,1094,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +1252,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +91,Lithuania,515,1,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +1241,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1023,Lithuania,455,1,Basic,Linear Algebra,Vector Spaces,Basis +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +511,Portugal,793,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +1288,Portugal,457,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +455,Portugal,545,1,Basic,Integration,Double Integration,"y-simple region (type I),Area ,Cartesian coordinates" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +61,Portugal,160,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1044,Lithuania,999,1,Basic,Differentiation,Partial Differentiation,First order +511,Portugal,239,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +168,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,287,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1288,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,454,1,Basic,Linear Algebra,Vector Spaces,Basis +28,Portugal,703,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Scalar product" +128,Lithuania,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1271,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +174,Portugal,721,0,Basic,Differentiation,Derivatives,"Quotient rule,Trigonometric rules" +204,Lithuania,798,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +769,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +414,Portugal,92,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +128,Lithuania,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +167,Portugal,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1173,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +376,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +669,Portugal,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +1229,Portugal,422,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +128,Lithuania,644,0,Basic,Linear Algebra,Matrices and Determinants,"Commuting matrices,Solve matrix equation,Matrix equivalent - echelon,Determinant computation,Matrix rank,Inverse matrix ,Determinant properties,Hermitian matrix,Matrix multiplication,Linearly independent rows,Symmetric matrix,Matrix operations" +986,Portugal,440,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Subspace,Linear combination,Dimension" +967,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +855,Italy,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1505,Portugal,1009,0,Basic,Optimization,Linear Optimization,"Transportation problem,Excel solver add-in,Simplex method,Linear programming" +986,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1158,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +988,Portugal,227,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +131,Lithuania,469,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +71,Portugal,334,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +415,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +1220,Slovenia,425,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +979,Portugal,1105,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +639,Italy,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +1565,Portugal,685,0,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +967,Portugal,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +851,Italy,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +852,Italy,227,0,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1080,Portugal,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1219,Portugal,437,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +985,Romania,796,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +972,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +591,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +375,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1247,Portugal,647,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations" +406,Portugal,243,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1195,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1220,Slovenia,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +731,Portugal,177,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +411,Portugal,336,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Modulus of a complex number,Imaginary part" +176,Italy,253,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Trigonometric rules" +576,Ireland,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1023,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +128,Lithuania,637,1,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +1219,Portugal,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1538,Portugal,1088,0,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +81,Lithuania,553,0,Basic,Analytic Geometry,Analytic Geometry,"Orthogonality,Distance,Scalar product" +986,Portugal,783,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +451,Portugal,322,0,Advanced,Probability,Probability,"Conditional probability,Bayes' Theorem" +81,Lithuania,372,1,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph,Path,Shortest path" +913,Ireland,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1220,Slovenia,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +953,Lithuania,185,1,Basic,Statistics,Statistics,"Frequency,Quantile,Mean,Variance,Median,Mode" +175,Italy,335,1,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +352,Portugal,142,0,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1487,Portugal,337,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form,Operations with complex numbers,Modulus of a complex number" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +37,Portugal,1202,1,Basic,Integration,Definite Integrals,"Fundamental theorem of Calculus,Volume of revolution" +1087,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +311,Ireland,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +969,Romania,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +90,Lithuania,339,0,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +1146,Portugal,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1220,Slovenia,377,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1229,Portugal,375,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1322,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +415,Portugal,650,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +483,Portugal,358,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Conjugate number,Operations with complex numbers,Real part" +91,Lithuania,354,1,Advanced,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +483,Portugal,262,1,Basic,Differentiation,Partial Differentiation,First order +1220,Slovenia,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,421,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +36,Portugal,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1375,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +966,Portugal,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +972,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +91,Lithuania,446,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +972,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +552,Portugal,983,0,Advanced,Real Functions of a single variable,Limits and Continuity,"Limits,Indeterminate forms" +1081,Portugal,277,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +854,Italy,952,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +57,Portugal,179,0,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +986,Portugal,447,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +1292,Italy,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +407,Portugal,341,1,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +91,Lithuania,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1517,Portugal,620,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +91,Lithuania,464,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometrical figures" +663,Portugal,191,0,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +885,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +455,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +189,Portugal,735,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +175,Italy,1108,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Lagrange's interpolation" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1148,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1098,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +411,Portugal,330,0,Basic,Complex Numbers,Complex Numbers,"Conjugate number,Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number" +1504,Portugal,360,0,Basic,Complex Numbers,Complex Numbers,"Principal argument,Trigonometric form,Modulus of a complex number,Algebraic form" +411,Portugal,621,0,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Characteristic polynomial,Eigenvalue,Diagonalization,Eigenvector,Spectrum" +979,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +199,Lithuania,665,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +885,Italy,454,0,Basic,Linear Algebra,Vector Spaces,Basis +625,Russian Federation,796,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Logarithmic function,Exponential function,Simplify expressions" +918,Portugal,257,0,Basic,Differentiation,Partial Differentiation,"Gradient,First order" +1197,Slovenia,427,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1023,Lithuania,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +720,Portugal,211,1,Basic,Statistics,Statistics,Linear regression +61,Portugal,156,0,Basic,Differential Equations,Differential Equations,Linear differential equation +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +165,Portugal,661,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +851,Italy,1013,1,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Stationary point,Minimum,Maximum,Saddle point" +352,Portugal,252,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Power rule,Trigonometric rules" +1220,Slovenia,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +642,Italy,204,1,Basic,Statistics,Statistics,"Student distribution,Hypothesis testing" +483,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1147,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1044,Lithuania,886,0,Advanced,Differentiation,Partial Differentiation,"Second order,First order" +351,Portugal,685,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +598,Italy,236,0,Advanced,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +576,Ireland,576,0,Advanced,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +840,Italy,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,434,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1220,Slovenia,435,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +374,Portugal,616,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +509,Portugal,666,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +167,Portugal,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +470,Portugal,348,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Complex plane ,Real part" +174,Portugal,254,1,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +57,Portugal,1123,1,Basic,Integration,Integration Techniques,"Integration by parts,Substitution,Simple integration,Direct inspection" +376,Portugal,345,0,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +884,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +988,Portugal,420,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +1322,Italy,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +979,Portugal,1092,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +972,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +168,Portugal,324,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +394,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1324,Italy,443,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Range,Kernel,Isomorphism" +876,Italy,375,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1109,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1037,Lithuania,256,0,Basic,Differentiation,Partial Differentiation,First order +90,Lithuania,338,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation,Trigonometric form" +1428,Portugal,437,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Composition of linear applications" +1564,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +91,Lithuania,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +509,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1247,Portugal,948,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1049,Lithuania,261,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +1146,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +41,Portugal,569,0,Basic,Analytic Geometry,Analytic Geometry,"Locus,Distance,Scalar product,Cross product,Orthogonality,Collinearity" +1557,Portugal,923,0,Basic,Real Functions of a single variable,"Domain, Image and Graphics",Domain +1220,Slovenia,291,1,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +81,Lithuania,571,0,Basic,Analytic Geometry,Analytic Geometry,"Intersection,Cartesian equations of a plane" +1220,Slovenia,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +600,Italy,652,1,Basic,Linear Algebra,Matrices and Determinants,"Linearly independent rows,Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +37,Portugal,341,0,Basic,Complex Numbers,Complex Numbers,Operations with complex numbers +1300,Italy,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +524,Portugal,118,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +91,Lithuania,463,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polyhedrons,Geometric solids,Classification of geometric solids" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +898,Portugal,428,0,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +625,Russian Federation,515,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Geometrical figures,Quadrilaterals,Rhombus" +483,Portugal,461,1,Basic,Fundamental Mathematics,Elementary Geometry,Platonic solids +1481,Portugal,558,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +955,Lithuania,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +753,Italy,259,0,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +509,Portugal,948,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Square linear system,Solution of linear system,Undetermined solution of linear system,Consistent system" +351,Portugal,114,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +576,Ireland,570,1,Advanced,Analytic Geometry,Analytic Geometry,"Distance,Orthogonal projection,Orthogonality" +704,Portugal,179,1,Basic,Statistics,Statistics,"Sample,Population,Mean,Variance,Standard deviation,Median,Mode,Quantile" +91,Lithuania,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +854,Italy,245,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +969,Romania,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +168,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +689,Portugal,818,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Maximum,Stationary point,Second derivative test,Unconstrained optimization,Saddle point,Optimal solution" +1229,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +967,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +376,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +1481,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +1289,Italy,780,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +853,Italy,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1135,Portugal,306,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism,Injective linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,428,1,Basic,Linear Algebra,Vector Spaces,"Change-of-basis matrix,Span,Basis,Dimension" +1289,Italy,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +840,Italy,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1219,Portugal,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +91,Lithuania,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1220,Slovenia,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +28,Portugal,90,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +642,Italy,202,0,Basic,Statistics,Statistics,"Student distribution,Confidence interval,Point estimate,Hypothesis testing" +1220,Slovenia,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1308,Italy,382,1,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +636,Italy,183,0,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +956,Portugal,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +884,Italy,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1350,Portugal,426,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +1132,Portugal,293,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +37,Portugal,1031,1,Advanced,Differentiation,Partial Differentiation,First order +483,Portugal,441,0,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +1219,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1411,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +72,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +91,Lithuania,505,1,Basic,Fundamental Mathematics,Elementary Geometry,Triangles +165,Portugal,878,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +966,Portugal,776,0,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +69,Portugal,549,1,Basic,Analytic Geometry,Analytic Geometry,"Distance,Cross product,Orthogonality" +599,Italy,605,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Spectrum,Diagonalization,Characteristic polynomial" +1220,Slovenia,423,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +37,Portugal,268,1,Basic,Differentiation,Partial Differentiation,"First order,Second order" +723,Portugal,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +1340,Portugal,1103,1,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +422,Portugal,326,0,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +684,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +80,Lithuania,275,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Product rule,Trigonometric rules" +411,Portugal,852,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1241,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +128,Lithuania,328,0,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +1284,Italy,439,1,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +406,Portugal,795,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Square roots,Simplify expressions" +510,Portugal,521,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Regular polygon" +1163,Portugal,453,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1277,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1505,Portugal,285,0,Basic,Optimization,Linear Optimization,Linear programming +411,Portugal,137,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Matrix multiplication,Symmetric matrix,Linearly independent rows,Determinant computation,Commuting matrices,Determinant properties,Inverse matrix ,Solve matrix equation" +634,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +411,Portugal,241,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1087,Portugal,459,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1081,Portugal,323,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +419,Portugal,353,0,Basic,Complex Numbers,Complex Numbers,"Nth root,Polar representation,Operations with complex numbers,Principal argument,De Moivre formulas" +988,Portugal,89,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +1179,Portugal,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +972,Portugal,439,0,Advanced,Linear Algebra,Vector Spaces,"Span,Subspace,Linear combination" +1319,Italy,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1273,Portugal,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +375,Portugal,352,1,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,853,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Continuity,Limits" +1286,Italy,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +415,Portugal,128,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Hermitian matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Matrix equivalent - echelon" +751,Lithuania,558,0,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Cartesian equations of a plane,Orthogonality,Cartesian equations of a line" +972,Portugal,384,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +979,Portugal,1088,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +468,Portugal,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1197,Slovenia,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +168,Portugal,444,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1277,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,383,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1525,Portugal,766,0,Basic,Probability,Probability,"Event,Sample space" +91,Lithuania,326,1,Basic,Complex Numbers,Complex Numbers,"Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number,Real part" +1220,Slovenia,379,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +1220,Slovenia,784,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1543,Portugal,1098,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +899,Lithuania,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1196,Slovenia,433,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1561,Portugal,833,0,Advanced,Differential Equations,Differential Equations,Linear differential equation +41,Portugal,191,1,Basic,Statistics,Statistics,"Median,Frequency,Mode,Quantile,Population,Data type,Sample,Relative frequency,Mean,Standard deviation,Variance" +81,Lithuania,547,1,Basic,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality,Scalar product" +913,Ireland,446,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span,Change-of-basis matrix" +380,Portugal,865,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Minimum,Maximum,Stationary point" +451,Portugal,312,0,Advanced,Probability,Probability,"Axioms of probability,Event,Addition, multiplication, and total probability rules,Sample space" +965,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +855,Italy,219,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +890,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,288,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +166,Portugal,563,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +1044,Lithuania,876,1,Basic,Differentiation,Partial Differentiation,"Directional derivative,Gradient,First order,Second order" +91,Lithuania,277,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1075,Portugal,885,0,Basic,Differentiation,Partial Differentiation,"Second order,First order" +855,Italy,233,0,Basic,Linear Algebra,Linear Systems,"Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system,Inconsistent system" +626,Italy,717,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Quotient rule" +1080,Portugal,325,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1070,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +986,Portugal,419,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,785,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +470,Portugal,555,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality,Locus,Distance" +1220,Slovenia,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1220,Slovenia,287,1,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +91,Lithuania,511,1,Basic,Fundamental Mathematics,Elementary Geometry,"Triangle inequality,Triangles" +969,Romania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +562,Ireland,638,0,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +840,Italy,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +525,Portugal,504,1,Basic,Fundamental Mathematics,Elementary Geometry,"Symmetry,Geometric transformations" +1220,Slovenia,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +422,Portugal,633,0,Advanced,Linear Algebra,Matrices and Determinants,"Matrix equivalent - echelon,Matrix rank,Matrix operations,Hermitian matrix,Matrix multiplication,Symmetric matrix,Linearly independent rows,Commuting matrices,Determinant computation,Determinant properties,Inverse matrix ,Solve matrix equation" +81,Lithuania,560,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +986,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +988,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +986,Portugal,289,0,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +69,Portugal,560,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Orthogonality" +972,Portugal,784,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +36,Portugal,1127,1,Basic,Set Theory,Set Theory,"Union of sets,Venn diagram,Complement of a set" +1158,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1182,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +956,Portugal,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +986,Portugal,424,1,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +988,Portugal,230,1,Basic,Linear Algebra,Linear Systems,"Consistent system,Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1194,Portugal,324,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Linear application" +1087,Portugal,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1044,Lithuania,267,0,Basic,Differentiation,Partial Differentiation,"First order,Second order" +349,Portugal,250,1,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +509,Portugal,608,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1179,Portugal,292,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +69,Portugal,554,0,Basic,Analytic Geometry,Analytic Geometry,"Scalar product,Orthogonality,Orthogonal projection,Distance" +1220,Slovenia,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +64,Portugal,735,1,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules,Logarithmic rule" +852,Italy,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +406,Portugal,138,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Determinant computation,Determinant properties,Inverse matrix" +406,Portugal,646,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix operations,Commuting matrices,Solve matrix equation,Determinant computation,Inverse matrix ,Matrix rank,Determinant properties,Matrix multiplication,Symmetric matrix" +509,Portugal,949,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +651,Ireland,181,0,Basic,Statistics,Statistics,"Quantile,Frequency,Sample,Mean,Variance,Standard deviation,Median,Mode" +876,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +1132,Portugal,783,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Isomorphism" +128,Lithuania,635,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties,Inverse matrix ,Matrix operations" +72,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +219,Lithuania,787,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Simplify expressions,Logarithmic function" +168,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +142,Portugal,614,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +351,Portugal,727,0,Advanced,Differentiation,Derivatives,"Trigonometric rules,Product rule" +510,Portugal,225,0,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1253,Portugal,296,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +91,Lithuania,524,0,Basic,Fundamental Mathematics,Elementary Geometry,"Quadrilaterals,Rhombus" +1291,Italy,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +741,Ireland,193,0,Basic,Statistics,Statistics,"Standard deviation,Variance,Median,Frequency,Mode,Outliers,Quantile,Interquartile range,Population,Data type,Sample,Relative frequency,Mean" +1087,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +582,Lithuania,345,1,Advanced,Complex Numbers,Complex Numbers,"Polar representation,Trigonometric form,Modulus of a complex number,Operations with complex numbers,Algebraic form,Principal argument,De Moivre formulas" +64,Portugal,108,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,435,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +1288,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +969,Romania,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +204,Lithuania,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1220,Slovenia,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +406,Portugal,127,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Determinant computation,Determinant properties,Symmetric matrix,Inverse matrix ,Linearly independent rows,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank" +1517,Portugal,293,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +91,Lithuania,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +625,Russian Federation,800,0,Advanced,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities","Exponential function,Simplify expressions" +509,Portugal,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +966,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1241,Portugal,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +659,Portugal,80,0,Basic,Statistics,Statistics,"Population,Variance,Median,Standard deviation,Mode,Quantile,Sample,Mean" +851,Italy,1005,0,Basic,Optimization,Nonlinear Optimization,"Minimum,Stationary point,Maximum,Saddle point,Optimal solution" +380,Portugal,859,0,Basic,Optimization,Nonlinear Optimization,"Optimal solution,Second derivative test,Minimum,Maximum,Stationary point,Saddle point" +64,Portugal,397,1,Basic,Integration,Integration Techniques,Substitution +1402,Portugal,380,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +966,Portugal,378,0,Basic,Linear Algebra,Linear Transformations,"Invertible linear operator,Isomorphism,Injective linear application" +28,Portugal,951,1,Basic,Linear Algebra,Linear Systems,"Square linear system,Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system" +851,Italy,377,1,Basic,Linear Algebra,Linear Transformations,"Range,Kernel,Matrix of a linear transformation" +1220,Slovenia,313,1,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1087,Portugal,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +913,Ireland,1095,0,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +979,Portugal,1107,1,Basic,Numerical Methods,Numerical Methods,"Newton’s method,Nonlinear equation" +405,Portugal,118,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1557,Portugal,927,0,Advanced,Real Functions of a single variable,"Domain, Image and Graphics",Planar graph +1505,Portugal,283,0,Basic,Optimization,Linear Optimization,"Excel solver add-in,Simplex method,Linear programming" +411,Portugal,640,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant computation,Determinant properties" +37,Portugal,311,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +525,Portugal,364,1,Basic,Fundamental Mathematics,Elementary Geometry,"Geometric solids,Geometrical figures,Classification of geometric solids" +72,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,459,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +511,Portugal,220,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +483,Portugal,448,1,Advanced,Linear Algebra,Vector Spaces,Subspace +986,Portugal,451,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1244,Portugal,450,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +91,Lithuania,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +511,Portugal,611,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +510,Portugal,607,0,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Eigenvector" +986,Portugal,381,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +535,Portugal,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +483,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +633,Italy,425,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Change-of-basis matrix,Linear combination,Span,Basis" +548,Portugal,486,0,Basic,Probability,Probability,"Event,Addition, multiplication, and total probability rules,Sample space" +91,Lithuania,778,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +986,Portugal,289,1,Advanced,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1028,Lithuania,998,0,Basic,Differentiation,Partial Differentiation,First order +1284,Italy,325,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +525,Portugal,519,1,Basic,Fundamental Mathematics,Elementary Geometry,Polygons +1087,Portugal,452,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +633,Italy,638,1,Advanced,Linear Algebra,Matrices and Determinants,"Solve matrix equation,Matrix equivalent - echelon,Inverse matrix ,Determinant computation,Determinant properties,Matrix rank,Hermitian matrix,Matrix operations,Symmetric matrix,Matrix multiplication,Linearly independent rows,Commuting matrices" +1220,Slovenia,455,0,Basic,Linear Algebra,Vector Spaces,Basis +1077,Portugal,319,1,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +1517,Portugal,857,1,Advanced,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvalue,Diagonalization,Spectrum,Characteristic polynomial" +1427,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +913,Ireland,297,0,Basic,Probability,Probability,"Sample space,Addition, multiplication, and total probability rules,Event,Axioms of probability" +1375,Portugal,779,1,Advanced,Linear Algebra,Linear Transformations,"Rotation,Linear application" +899,Lithuania,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +851,Italy,155,0,Basic,Optimization,Nonlinear Optimization,"Lagrange multipliers,Maximum,Minimum" +663,Portugal,183,1,Basic,Statistics,Statistics,"Quantile,Population,Sample,Mean,Variance,Standard deviation,Median,Mode" +468,Portugal,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +1481,Portugal,298,0,Basic,Probability,Probability,"Event,Axioms of probability" +913,Ireland,487,0,Basic,Probability,Probability,"Axioms of probability,Event,Independence,Addition, multiplication, and total probability rules,Sample space" +1331,Portugal,431,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +80,Lithuania,666,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +973,Portugal,224,1,Basic,Linear Algebra,Linear Systems,"Inconsistent system,Solution of linear system,Undetermined solution of linear system,Consistent system,Square linear system" +1319,Italy,383,0,Basic,Linear Algebra,Linear Transformations,"Linearity,Matrix of a linear transformation,Linear application" +899,Lithuania,436,0,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +600,Italy,125,0,Basic,Linear Algebra,Matrices and Determinants,"Determinant properties,Symmetric matrix,Inverse matrix ,Commuting matrices,Solve matrix equation,Matrix operations,Matrix rank,Determinant computation,Matrix multiplication" +751,Lithuania,699,1,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Locus,Intersection,Cartesian equations of a line" +37,Portugal,401,1,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +1288,Portugal,292,0,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel,Isomorphism" +1220,Slovenia,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +376,Portugal,454,0,Basic,Linear Algebra,Vector Spaces,Basis +1322,Italy,421,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1023,Lithuania,452,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1044,Lithuania,882,0,Advanced,Differentiation,Partial Differentiation,"Gradient,First order,Directional derivative" +168,Portugal,313,0,Basic,Linear Algebra,Vector Spaces,"Span,Linear combination,Subspace" +1241,Portugal,422,0,Basic,Linear Algebra,Vector Spaces,"Linear combination,Span" +1518,Portugal,104,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +1080,Portugal,457,1,Basic,Linear Algebra,Vector Spaces,"Linear combination,Subspace,Span" +1288,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1308,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +625,Russian Federation,458,1,Basic,Fundamental Mathematics,Elementary Geometry,"Polygons,Geometric transformations,Geometrical figures" +91,Lithuania,384,1,Basic,Linear Algebra,Linear Transformations,"Linearity,Invertible linear operator,Composition of linear applications,Linear application" +884,Italy,321,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +483,Portugal,254,0,Basic,Differentiation,Derivatives,"Logarithmic rule,Power rule,Trigonometric rules" +41,Portugal,681,1,Basic,Real Functions of a single variable,Limits and Continuity,"Indeterminate forms,Limits" +1220,Slovenia,296,1,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +72,Portugal,438,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +177,Italy,798,1,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +913,Ireland,761,1,Basic,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +633,Italy,420,1,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +72,Portugal,334,0,Basic,Complex Numbers,Complex Numbers,"Algebraic form,Principal argument,De Moivre formulas,Polar representation,Operations with complex numbers,Modulus of a complex number,Trigonometric form" +958,Lithuania,423,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +403,Portugal,782,0,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1291,Italy,450,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,317,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +1220,Slovenia,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1247,Portugal,347,0,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +85,Lithuania,1141,1,Basic,Set Theory,Set Theory,Subset +972,Portugal,436,1,Basic,Linear Algebra,Vector Spaces,"Euclidean spaces,Orthogonal basis,Basis" +509,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1132,Portugal,278,1,Advanced,Linear Algebra,Linear Transformations,Matrix of a linear transformation +899,Lithuania,317,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Span,Linear combination" +37,Portugal,1214,1,Basic,Numerical Methods,Numerical Methods,"Mean,Median,Outliers,Interquartile range" +37,Portugal,142,1,Basic,Differentiation,Derivatives,"Trigonometric rules,Product rule" +1292,Italy,593,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Spectrum,Characteristic polynomial,Eigenvalue" +1023,Lithuania,433,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Dimension" +155,Spain,168,1,Basic,Differential Equations,Differential Equations,Linear differential equation +1411,Portugal,288,1,Advanced,Linear Algebra,Linear Transformations,"Range,Kernel" +884,Italy,319,0,Basic,Linear Algebra,Linear Transformations,"Matrix of a linear transformation,Linear application,Linearity" +955,Lithuania,417,0,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +81,Lithuania,365,0,Advanced,Graph Theory,Graph Theory,"Adjacency matrix,Simple graph" +1517,Portugal,382,0,Advanced,Linear Algebra,Linear Transformations,"Injective linear application,Invertible linear operator,Isomorphism" +1220,Slovenia,444,0,Basic,Linear Algebra,Linear Transformations,"Range,Kernel" +1340,Portugal,1113,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +972,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +753,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +913,Ireland,188,0,Basic,Statistics,Statistics,"Mode,Quantile,Population,Data type,Sample,Mean,Standard deviation,Variance,Median" +855,Italy,91,0,Basic,Linear Algebra,Linear Systems,"Consistent system,Inconsistent system,Solution of linear system,Undetermined solution of linear system" +1081,Portugal,424,0,Advanced,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1543,Portugal,1101,0,Basic,Numerical Methods,Numerical Methods,"Polynomial interpolation,Newton’s interpolation" +351,Portugal,120,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +535,Portugal,429,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +1146,Portugal,429,0,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +979,Portugal,1251,1,Basic,Numerical Methods,Numerical Methods,"Newton’s interpolation,Polynomial interpolation" +509,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +1244,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +884,Italy,291,0,Basic,Linear Algebra,Linear Transformations,"Kernel,Isomorphism,Injective linear application,Range" +1525,Portugal,775,0,Advanced,Probability,Probability,"Event,Sample space,Addition, multiplication, and total probability rules,Axioms of probability" +1242,Portugal,454,1,Basic,Linear Algebra,Vector Spaces,Basis +890,Portugal,376,0,Basic,Linear Algebra,Linear Transformations,Matrix of a linear transformation +1079,Portugal,431,1,Basic,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +208,Lithuania,645,1,Basic,Linear Algebra,Matrices and Determinants,"Matrix rank,Linearly independent rows" +1220,Slovenia,417,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +576,Ireland,550,0,Advanced,Analytic Geometry,Analytic Geometry,"Cross product,Orthogonality" +1517,Portugal,430,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Linear combination,Span" +890,Portugal,441,1,Advanced,Linear Algebra,Vector Spaces,"Basis,Span" +37,Portugal,596,1,Basic,Linear Algebra,Eigenvalues and Eigenvectors,"Eigenvector,Eigenvalue" +177,Italy,250,0,Basic,Differentiation,Derivatives,"Quotient rule,Product rule,Trigonometric rules" +174,Portugal,328,1,Basic,Complex Numbers,Complex Numbers,"Real part,Operations with complex numbers,Imaginary part,Modulus of a complex number,Conjugate number" +72,Portugal,427,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Basis,Dimension" +406,Portugal,667,0,Basic,Fundamental Mathematics,"Algebraic expressions, Equations, and Inequalities",Simplify expressions +239,Lithuania,93,0,Basic,Differentiation,Derivatives,"Product rule,Logarithmic rule,Trigonometric rules" +955,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +71,Portugal,651,0,Basic,Linear Algebra,Matrices and Determinants,"Matrix multiplication,Symmetric matrix,Matrix operations,Commuting matrices,Solve matrix equation,Inverse matrix ,Determinant computation,Matrix rank,Determinant properties" +91,Lithuania,419,0,Advanced,Linear Algebra,Vector Spaces,"Subspace,Dimension" +1026,Lithuania,658,0,Basic,Differentiation,Derivatives,"Product rule,Trigonometric rules" +411,Portugal,393,0,Basic,Integration,Integration Techniques,"Direct inspection,Simple integration" +972,Portugal,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +840,Italy,780,1,Basic,Linear Algebra,Linear Transformations,"Injective linear application,Range,Kernel,Isomorphism" +374,Portugal,415,0,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1340,Portugal,1089,1,Basic,Numerical Methods,Numerical Methods,"First order,Second order" +1044,Lithuania,261,1,Basic,Differentiation,Partial Differentiation,"First order,Gradient" +535,Portugal,346,1,Advanced,Complex Numbers,Complex Numbers,"Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part" +37,Portugal,339,1,Basic,Complex Numbers,Complex Numbers,"Trigonometric form,Operations with complex numbers,Modulus of a complex number,Algebraic form,Principal argument,De Moivre formulas,Polar representation" +483,Portugal,352,0,Advanced,Complex Numbers,Complex Numbers,"Conjugate number,Operations with complex numbers" +91,Lithuania,498,0,Advanced,Fundamental Mathematics,Elementary Geometry,"Triangles,Classification of geometrical figures,Geometrical figures" +1220,Slovenia,416,1,Basic,Linear Algebra,Vector Spaces,"Basis,Dimension" +1220,Slovenia,323,0,Basic,Linear Algebra,Linear Transformations,"Linear application,Linearity" +576,Ireland,561,0,Basic,Analytic Geometry,Analytic Geometry,"Cartesian equations of a plane,Cartesian equations of a line" +91,Lithuania,453,1,Basic,Linear Algebra,Vector Spaces,"Linear independence,Span,Linear dependence" +1517,Portugal,448,0,Advanced,Linear Algebra,Vector Spaces,Subspace diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/adapter_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/adapter_report.json new file mode 100644 index 0000000000000000000000000000000000000000..28202c12ad7086708196ee1872c58b0906fe251b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/adapter_report.json @@ -0,0 +1,7 @@ +{ + "adapter_ready_status": "pass", + "adapter_fail_reason_code": null, + "adapter_fail_detail": null, + "adapter_transforms_applied": [], + "model_input_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/model_input_manifest.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/adapter_transforms_applied.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/adapter_transforms_applied.json new file mode 100644 index 0000000000000000000000000000000000000000..0637a088a01e8ddab3bf3fa98dbe804cbde1a0dc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/adapter_transforms_applied.json @@ -0,0 +1 @@ +[] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/model_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..fffa47cdb0ec3d4943bcefaad60631b895ba321d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/tvae/model_input_manifest.json @@ -0,0 +1,176 @@ +{ + "dataset_id": "c6", + "model": "tvae", + "target_column": "Type of Answer", + "task_type": "classification", + "column_schema": [ + { + "name": "Student ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 367, + "unique_ratio": 0.048062, + "example_values": [ + "473", + "351", + "967", + "1557", + "394" + ] + } + }, + { + "name": "Student Country", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 8, + "unique_ratio": 0.001048, + "example_values": [ + "Portugal", + "Italy", + "Lithuania", + "Slovenia", + "Ireland" + ] + } + }, + { + "name": "Question ID", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 796, + "unique_ratio": 0.104243, + "example_values": [ + "346", + "796", + "453", + "87", + "325" + ] + } + }, + { + "name": "Type of Answer", + "role": "target", + "semantic_type": "boolean", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "0", + "1" + ] + } + }, + { + "name": "Question Level", + "role": "feature", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000262, + "example_values": [ + "Advanced", + "Basic" + ] + } + }, + { + "name": "Topic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14, + "unique_ratio": 0.001833, + "example_values": [ + "Complex Numbers", + "Fundamental Mathematics", + "Linear Algebra", + "Real Functions of a single variable", + "Analytic Geometry" + ] + } + }, + { + "name": "Subtopic", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 24, + "unique_ratio": 0.003143, + "example_values": [ + "Complex Numbers", + "Algebraic expressions, Equations, and Inequalities", + "Vector Spaces", + "Limits and Continuity", + "Linear Transformations" + ] + } + }, + { + "name": "Keywords", + "role": "feature", + "semantic_type": "text", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "keep_raw", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 360, + "unique_ratio": 0.047145, + "example_values": [ + "Imaginary part,Modulus of a complex number,Operations with complex numbers,Conjugate number,Real part", + "Logarithmic function,Exponential function,Simplify expressions", + "Linear independence,Span,Linear dependence", + "Indeterminate forms,Limits", + "Range,Kernel" + ] + } + } + ], + "public_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/public_gate/staged_input_manifest.json", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae_metadata.json b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae_metadata.json new file mode 100644 index 0000000000000000000000000000000000000000..fd44f1730c28205b8349695531f19bedb2470087 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/c6/tvae/tvae-c6-20260419_191329/tvae_metadata.json @@ -0,0 +1,36 @@ +{ + "columns": [ + { + "name": "Student ID", + "type": "continuous" + }, + { + "name": "Student Country", + "type": "categorical" + }, + { + "name": "Question ID", + "type": "continuous" + }, + { + "name": "Type of Answer", + "type": "categorical" + }, + { + "name": "Question Level", + "type": "categorical" + }, + { + "name": "Topic", + "type": "categorical" + }, + { + "name": "Subtopic", + "type": "categorical" + }, + { + "name": "Keywords", + "type": "categorical" + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/_bayesnet_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/_bayesnet_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..5e0d9c39b11f92a38c684645752cc5cd91e45b4e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/_bayesnet_generate.py @@ -0,0 +1,43 @@ +import subprocess, sys, os + +pip_libs = "/pip_libs" +sys.path.insert(0, pip_libs) +os.environ["PYTHONPATH"] = pip_libs + os.pathsep + os.environ.get("PYTHONPATH", "") + +def _ensure_deps(): + try: + import synthcity + except ModuleNotFoundError: + print("[BayesNet] synthcity not found - installing to cache...") + subprocess.run( + [sys.executable, "-m", "pip", "install", + "--target", pip_libs, "synthcity==0.2.12", "numpy<2", "-q"], + check=True + ) + import shutil, glob + for pat in ["torch", "torch-*", "torchvision", "torchvision-*", + "torchvision.libs", "torchgen", "nvidia*", "triton*"]: + for p in glob.glob(os.path.join(pip_libs, pat)): + if os.path.isdir(p): shutil.rmtree(p) + else: os.remove(p) + if pip_libs not in sys.path: + sys.path.insert(0, pip_libs) + +_ensure_deps() + +import pickle, json as _json +with open("/work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet_model.pkl", "rb") as f: + plugin = pickle.load(f) +syn = plugin.generate(count=15215).dataframe() + +# Restore zero-variance columns that were dropped during training +const_path = "/work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet_model.pkl".replace("bayesnet_model.pkl", "const_cols.json") +if os.path.exists(const_path): + with open(const_path) as _f: + const_cols = _json.load(_f) + for col, val in const_cols.items(): + syn[col] = val + print(f"[BayesNet] Restored constant column '{col}' = {val}") + +syn.to_csv("/work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet-n11-15215-20260330_070515.csv", index=False) +print(f"[BayesNet] Generated 15215 rows -> /work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet-n11-15215-20260330_070515.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/_bayesnet_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/_bayesnet_train.py new file mode 100644 index 0000000000000000000000000000000000000000..dd2b75263f681de02d45aa09335ec78f277f6844 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/_bayesnet_train.py @@ -0,0 +1,62 @@ +import subprocess, sys, os + +pip_libs = "/pip_libs" +sys.path.insert(0, pip_libs) +os.environ["PYTHONPATH"] = pip_libs + os.pathsep + os.environ.get("PYTHONPATH", "") + +def _ensure_deps(): + try: + import synthcity + except ModuleNotFoundError: + print("[BayesNet] synthcity not found - installing to cache (first run, may take minutes)...") + # Install synthcity with numpy<2 to avoid conflicts + subprocess.run( + [sys.executable, "-m", "pip", "install", + "--target", pip_libs, "synthcity==0.2.12", "numpy<2", "-q"], + check=True + ) + # Remove torch/torchvision from pip_libs to avoid shadowing system versions + import shutil, glob + for pat in ["torch", "torch-*", "torchvision", "torchvision-*", + "torchvision.libs", "torchgen", "nvidia*", "triton*"]: + for p in glob.glob(os.path.join(pip_libs, pat)): + if os.path.isdir(p): shutil.rmtree(p) + else: os.remove(p) + if pip_libs not in sys.path: + sys.path.insert(0, pip_libs) + +_ensure_deps() + +from synthcity.plugins import Plugins +import pickle +import pandas as pd +from synthcity.plugins.core.dataloader import GenericDataLoader + +df = pd.read_csv("/work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/staged/public/train.csv") +df = df.dropna(axis=1, how="all") + +# Drop zero-variance columns (only 1 unique value) to avoid +# synthcity encoder KeyError during generation +import json as _json +const_cols = {} +for col in list(df.columns): + nuniq = df[col].nunique() + if nuniq <= 1: + const_cols[col] = df[col].iloc[0] if len(df) > 0 else None + df = df.drop(columns=[col]) + print(f"[BayesNet] Dropped zero-variance column '{col}' (value={const_cols[col]})") + +# Save constant columns info so generate can restore them +const_path = "/work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet_model.pkl".replace("bayesnet_model.pkl", "const_cols.json") +with open(const_path, "w") as _f: + _json.dump({k: str(v) for k, v in const_cols.items()}, _f) + +print(f"[BayesNet] Training on {len(df)} rows, {len(df.columns)} cols") + +loader = GenericDataLoader(df) +plugin = Plugins().get("bayesian_network") +plugin.fit(loader) + +with open("/work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet_model.pkl", "wb") as f: + pickle.dump(plugin, f) +print(f"[BayesNet] Model saved -> /work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet_model.pkl") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/gen_20260321_085714.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/gen_20260321_085714.log new file mode 100644 index 0000000000000000000000000000000000000000..84b54497a4c0d3afee5fde7fdb87ccc0c0cc2c07 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/gen_20260321_085714.log @@ -0,0 +1,3 @@ +03/21/2026 00:58:20:WARNING:Probability values don't exactly sum to 1. Differ by: -2.220446049250313e-16. Adjusting values. +[KeOps] Warning : CUDA libraries not found or could not be loaded; Switching to CPU only. +[BayesNet] Generated 1000 rows -> /work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet-n11-1000-20260321_085714.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/gen_20260330_070515.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/gen_20260330_070515.log new file mode 100644 index 0000000000000000000000000000000000000000..4ec9d367c01dbeeac2b305e6ff07e19c4535ad4b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/gen_20260330_070515.log @@ -0,0 +1,3 @@ +03/29/2026 23:06:52:WARNING:Probability values don't exactly sum to 1. Differ by: -2.220446049250313e-16. Adjusting values. +[KeOps] Warning : CUDA libraries not found or could not be loaded; Switching to CPU only. +[BayesNet] Generated 15215 rows -> /work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet-n11-15215-20260330_070515.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..6c359aa8e271fbf6fdec18331c2f91421b80cc91 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n11", + "model": "bayesnet", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-train.csv", + "exists": true, + "size": 1197106, + "sha256": "f8253c0573a8f028c0b1e17752d214fbad956d0ace350d0dee3947f3dfafbc75" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-val.csv", + "exists": true, + "size": 149602, + "sha256": "7bde20178eb7de9523d4df719a3b461656f5704e3a03187c8289b2299dc8e458" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-test.csv", + "exists": true, + "size": 149859, + "sha256": "855b55969de2da1341d428edef6dedfdf0c60f016b9fa08efaf8a883af6a2c53" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_profile.json", + "exists": true, + "size": 5104, + "sha256": "0bd368d22598246defd33bda12c9872165f26e798cc462ec472d193f322fba07" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_contract_v1.json", + "exists": true, + "size": 5874, + "sha256": 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+[BayesNet] Training on 15215 rows, 11 cols +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and binary values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +/pip_libs/sklearn/metrics/cluster/_supervised.py:58: UserWarning: Clustering metrics expects discrete values but received continuous values for label, and continuous values for target + warnings.warn(msg, UserWarning) +[BayesNet] Model saved -> /work/output-SpecializedModels/n11/bayesnet/bayesnet-n11-20260321_085202/bayesnet_model.pkl diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/ctgan_metadata.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/ctgan_metadata.json new file mode 100644 index 0000000000000000000000000000000000000000..cb19699b85fe49965d86e54abd3c67a8ec6163a8 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/ctgan_metadata.json @@ -0,0 +1,48 @@ +{ + "columns": [ + { + "name": "28.7967", + "type": "continuous" + }, + { + "name": "16.0021", + "type": "continuous" + }, + { + "name": "2.6449", + "type": "continuous" + }, + { + "name": "0.3918", + "type": "continuous" + }, + { + "name": "0.1982", + "type": "continuous" + }, + { + "name": "27.7004", + "type": "continuous" + }, + { + "name": "22.011", + "type": "continuous" + }, + { + "name": "-8.2027", + "type": "continuous" + }, + { + "name": "40.092", + "type": "continuous" + }, + { + "name": "81.8828", + "type": "continuous" + }, + { + "name": "g", + "type": "categorical" + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/gen_20260328_102858.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/gen_20260328_102858.log new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/gen_20260330_070426.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/gen_20260330_070426.log new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..d0df46d6279ceed752adf467715dbe95c309efe1 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n11", + "model": "ctgan", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-train.csv", + "exists": true, + "size": 1197106, + "sha256": "f8253c0573a8f028c0b1e17752d214fbad956d0ace350d0dee3947f3dfafbc75" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-val.csv", + "exists": true, + "size": 149602, + "sha256": "7bde20178eb7de9523d4df719a3b461656f5704e3a03187c8289b2299dc8e458" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-test.csv", + "exists": true, + "size": 149859, + "sha256": "855b55969de2da1341d428edef6dedfdf0c60f016b9fa08efaf8a883af6a2c53" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_profile.json", + "exists": true, + "size": 5104, + "sha256": "0bd368d22598246defd33bda12c9872165f26e798cc462ec472d193f322fba07" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_contract_v1.json", + "exists": true, + "size": 5874, + "sha256": "ad3f01eafe9fb37754eed9be9bd058390fd3a4f54d8be4ceb4b5326a9901da9d" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/models_300epochs/train_20260328_054002.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/models_300epochs/train_20260328_054002.log new file mode 100644 index 0000000000000000000000000000000000000000..19673a95552befef13608269dff934aa13d5cf14 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/models_300epochs/train_20260328_054002.log @@ -0,0 +1,2 @@ +/opt/conda/lib/python3.10/site-packages/torch/autograd/graph.py:841: UserWarning: Attempting to run cuBLAS, but there was no current CUDA context! Attempting to set the primary context... (Triggered internally at /pytorch/aten/src/ATen/cuda/CublasHandlePool.cpp:270.) + return Variable._execution_engine.run_backward( # Calls into the C++ engine to run the backward pass diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..96904695eb041844b615aa285fda674f1da87108 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/ctgan/ctgan-n11-20260328_054001/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,235 @@ +{ + "dataset_id": "n11", + "target_column": "g", + "task_type": "classification", + "columns": [ + { + "name": "28.7967", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + 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[], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14824, + "unique_ratio": 0.974302, + "example_values": [ + "132.907", + "286.69", + "108.464", + "180.4168", + "222.525" + ] + } + }, + { + "name": "g", + "role": "target", + "semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000131, + "example_values": [ + "g", + "h" + ] + } + } + ], + "public_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/public_gate/staged_input_manifest.json", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/train_20260331_113635.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/train_20260331_113635.log new file mode 100644 index 0000000000000000000000000000000000000000..82e00f6a490d58ba52a35c4146489d2248752d12 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/train_20260331_113635.log @@ -0,0 +1,53129 @@ + +Map: 0%| | 0/15215 [00:00 /work/output-SpecializedModels/n11/realtabformer/rtf-n11-20260331_113634/models_100epochs diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/_tabddpm_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/_tabddpm_train.py new file mode 100644 index 0000000000000000000000000000000000000000..ea5ddb12e96d2dc3650dde6c37c4e58a81d4f7ef --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/_tabddpm_train.py @@ -0,0 +1,32 @@ +import os, sys, subprocess + +tabddpm_root = "/workspace/tabddpm/code" +assert os.path.isdir(tabddpm_root), f"TabDDPM source not mounted: {tabddpm_root}" +env = os.environ.copy() +env["PYTHONPATH"] = tabddpm_root + (os.pathsep + env.get("PYTHONPATH", "")) + +# Write a wrapper that patches collections.Sequence (removed in Python 3.10+) +# before running pipeline.py - needed because skorch uses old API +wrapper = os.path.join(tabddpm_root, "_compat_run.py") +with open(wrapper, "w") as f: + f.write( + "import collections, collections.abc\n" + "for _a in ('Sequence','MutableSequence','MutableMapping','Mapping'," + "'MutableSet','Set','Callable','Iterable','Iterator'):\n" + " if not hasattr(collections, _a): setattr(collections, _a, getattr(collections.abc, _a, None))\n" + "import sys, runpy\n" + "sys.argv = sys.argv[1:]\n" + "runpy.run_path(sys.argv[0], run_name='__main__')\n" + ) + +print(f"[TabDDPM] Training, config=/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/config.toml") +ret = subprocess.run( + [sys.executable, wrapper, "scripts/pipeline.py", + "--config", "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/config.toml", + "--train"], + cwd=tabddpm_root, + env=env +) +if ret.returncode != 0: + sys.exit(ret.returncode) +print("[TabDDPM] Training complete") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/config.toml b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/config.toml new file mode 100644 index 0000000000000000000000000000000000000000..ad681730a405f17a1a08437dc8b419bec2ff614d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/config.toml @@ -0,0 +1,39 @@ +seed = 0 +parent_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/output" +real_data_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data" +model_type = "mlp" +num_numerical_features = 10 +device = "cuda:0" + +[model_params] +d_in = 10 +num_classes = 2 +is_y_cond = true + +[model_params.rtdl_params] +d_layers = [256, 256] +dropout = 0.0 + +[diffusion_params] +num_timesteps = 1000 +gaussian_loss_type = "mse" + +[train.main] +steps = 5000 +lr = 0.001 +weight_decay = 0.0 +batch_size = 256 + +[train.T] +seed = 0 +normalization = "quantile" +num_nan_policy = "__none__" +cat_nan_policy = "__none__" +cat_min_frequency = "__none__" +cat_encoding = "__none__" +y_policy = "default" + +[sample] +num_samples = 1000 +batch_size = 1000 +seed = 0 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/X_num_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/X_num_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..0ce6a0d07aaef7146f6fe7039c7d1215095e58a0 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/X_num_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/X_num_val.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/X_num_val.npy new file mode 100644 index 0000000000000000000000000000000000000000..06e08b511f502f84c134dcb9dd268fd9dfb4c127 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/X_num_val.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/info.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/info.json new file mode 100644 index 0000000000000000000000000000000000000000..ad93acde945d46916d99f9a52fb19bb740a3d107 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/info.json @@ -0,0 +1,37 @@ +{ + "name": "benchmark_dataset", + "task_type": "multiclass", + "n_num_features": 10, + "n_cat_features": 0, + "train_size": 15215, + "num_col_idx": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9 + ], + "cat_col_idx": [], + "target_col_idx": [ + 10 + ], + "column_names": [ + "28.7967", + "16.0021", + "2.6449", + "0.3918", + "0.1982", + "27.7004", + "22.011", + "-8.2027", + "40.092", + "81.8828", + "g" + ], + "num_classes": 2 +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..feb929f859918c63836a85fc611682370914ad27 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_train.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_train.npy new file mode 100644 index 0000000000000000000000000000000000000000..c9468fb143fb4c6783275c5964196d9f8c85055b Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_train.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_val.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_val.npy new file mode 100644 index 0000000000000000000000000000000000000000..6ce133837f0c3202112843f78a8daebbaa0e1f64 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/data/y_val.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/train_20260319_074810.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/train_20260319_074810.log new file mode 100644 index 0000000000000000000000000000000000000000..96ccb56c8d0f814f14f781809cdab9b2d7de6c22 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/train_20260319_074810.log @@ -0,0 +1,26 @@ +/opt/conda/lib/python3.11/site-packages/skorch/__init__.py:6: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81. + import pkg_resources +[0] +10 +{'d_in': 10, 'num_classes': 2, 'is_y_cond': True, 'rtdl_params': {'d_layers': [256, 256], 'dropout': 0.0}} +mlp +Traceback (most recent call last): + File "/workspace/tabddpm/code/_compat_run.py", line 6, in + runpy.run_path(sys.argv[0], run_name='__main__') + File "", line 291, in run_path + File "", line 98, in _run_module_code + File "", line 88, in _run_code + File "scripts/pipeline.py", line 112, in + main() + File "scripts/pipeline.py", line 48, in main + train( + File "/workspace/tabddpm/code/scripts/train.py", line 154, in train + trainer.run_loop() + File "/workspace/tabddpm/code/scripts/train.py", line 53, in run_loop + x, out_dict = next(self.train_iter) + ^^^^^^^^^^^^^^^^^^^^^ + File "/workspace/tabddpm/code/lib/data.py", line 605, in prepare_fast_dataloader + y = torch.from_numpy(D.y[split]) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +TypeError: can't convert np.ndarray of type numpy.object_. The only supported types are: float64, float32, float16, complex64, complex128, int64, int32, int16, int8, uint64, uint32, uint16, uint8, and bool. +[TabDDPM] Training, config=/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_074807/config.toml diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/_tabddpm_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/_tabddpm_train.py new file mode 100644 index 0000000000000000000000000000000000000000..983f6d6be59b4bad3ad8fe71476459dc67217305 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/_tabddpm_train.py @@ -0,0 +1,32 @@ +import os, sys, subprocess + +tabddpm_root = "/workspace/tabddpm/code" +assert os.path.isdir(tabddpm_root), f"TabDDPM source not mounted: {tabddpm_root}" +env = os.environ.copy() +env["PYTHONPATH"] = tabddpm_root + (os.pathsep + env.get("PYTHONPATH", "")) + +# Write a wrapper that patches collections.Sequence (removed in Python 3.10+) +# before running pipeline.py - needed because skorch uses old API +wrapper = os.path.join(tabddpm_root, "_compat_run.py") +with open(wrapper, "w") as f: + f.write( + "import collections, collections.abc\n" + "for _a in ('Sequence','MutableSequence','MutableMapping','Mapping'," + "'MutableSet','Set','Callable','Iterable','Iterator'):\n" + " if not hasattr(collections, _a): setattr(collections, _a, getattr(collections.abc, _a, None))\n" + "import sys, runpy\n" + "sys.argv = sys.argv[1:]\n" + "runpy.run_path(sys.argv[0], run_name='__main__')\n" + ) + +print(f"[TabDDPM] Training, config=/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/config.toml") +ret = subprocess.run( + [sys.executable, wrapper, "scripts/pipeline.py", + "--config", "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/config.toml", + "--train"], + cwd=tabddpm_root, + env=env +) +if ret.returncode != 0: + sys.exit(ret.returncode) +print("[TabDDPM] Training complete") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/config.toml b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/config.toml new file mode 100644 index 0000000000000000000000000000000000000000..6d1206fb92bff7bdb7d4f745554f604bb8bdbd89 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/config.toml @@ -0,0 +1,39 @@ +seed = 0 +parent_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/output" +real_data_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data" +model_type = "mlp" +num_numerical_features = 10 +device = "cuda:0" + +[model_params] +d_in = 10 +num_classes = 2 +is_y_cond = true + +[model_params.rtdl_params] +d_layers = [256, 256] +dropout = 0.0 + +[diffusion_params] +num_timesteps = 1000 +gaussian_loss_type = "mse" + +[train.main] +steps = 5000 +lr = 0.001 +weight_decay = 0.0 +batch_size = 256 + +[train.T] +seed = 0 +normalization = "quantile" +num_nan_policy = "__none__" +cat_nan_policy = "__none__" +cat_min_frequency = "__none__" +cat_encoding = "__none__" +y_policy = "default" + +[sample] +num_samples = 1000 +batch_size = 1000 +seed = 0 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/X_num_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/X_num_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..0ce6a0d07aaef7146f6fe7039c7d1215095e58a0 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/X_num_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/X_num_val.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/X_num_val.npy new file mode 100644 index 0000000000000000000000000000000000000000..06e08b511f502f84c134dcb9dd268fd9dfb4c127 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/X_num_val.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/info.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/info.json new file mode 100644 index 0000000000000000000000000000000000000000..ad93acde945d46916d99f9a52fb19bb740a3d107 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/info.json @@ -0,0 +1,37 @@ +{ + "name": "benchmark_dataset", + "task_type": "multiclass", + "n_num_features": 10, + "n_cat_features": 0, + "train_size": 15215, + "num_col_idx": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9 + ], + "cat_col_idx": [], + "target_col_idx": [ + 10 + ], + "column_names": [ + "28.7967", + "16.0021", + "2.6449", + "0.3918", + "0.1982", + "27.7004", + "22.011", + "-8.2027", + "40.092", + "81.8828", + "g" + ], + "num_classes": 2 +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..feb929f859918c63836a85fc611682370914ad27 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_train.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_train.npy new file mode 100644 index 0000000000000000000000000000000000000000..c9468fb143fb4c6783275c5964196d9f8c85055b Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_train.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_val.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_val.npy new file mode 100644 index 0000000000000000000000000000000000000000..6ce133837f0c3202112843f78a8daebbaa0e1f64 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/data/y_val.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/train_20260319_225956.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/train_20260319_225956.log new file mode 100644 index 0000000000000000000000000000000000000000..6c283065756111cd78b3b38c84db3a74ff473347 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/train_20260319_225956.log @@ -0,0 +1,26 @@ +/opt/conda/lib/python3.11/site-packages/skorch/__init__.py:6: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81. + import pkg_resources +[0] +10 +{'d_in': 10, 'num_classes': 2, 'is_y_cond': True, 'rtdl_params': {'d_layers': [256, 256], 'dropout': 0.0}} +mlp +Traceback (most recent call last): + File "/workspace/tabddpm/code/_compat_run.py", line 6, in + runpy.run_path(sys.argv[0], run_name='__main__') + File "", line 291, in run_path + File "", line 98, in _run_module_code + File "", line 88, in _run_code + File "scripts/pipeline.py", line 112, in + main() + File "scripts/pipeline.py", line 48, in main + train( + File "/workspace/tabddpm/code/scripts/train.py", line 154, in train + trainer.run_loop() + File "/workspace/tabddpm/code/scripts/train.py", line 53, in run_loop + x, out_dict = next(self.train_iter) + ^^^^^^^^^^^^^^^^^^^^^ + File "/workspace/tabddpm/code/lib/data.py", line 605, in prepare_fast_dataloader + y = torch.from_numpy(D.y[split]) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +TypeError: can't convert np.ndarray of type numpy.object_. The only supported types are: float64, float32, float16, complex64, complex128, int64, int32, int16, int8, uint64, uint32, uint16, uint8, and bool. +[TabDDPM] Training, config=/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260319_225953/config.toml diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_sample.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_sample.py new file mode 100644 index 0000000000000000000000000000000000000000..d11ab2c671ff195623f55fe34902c47643f23050 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_sample.py @@ -0,0 +1,66 @@ +import os, sys, subprocess, json +import numpy as np +import pandas as pd + +tabddpm_root = "/workspace/tabddpm/code" +assert os.path.isdir(tabddpm_root), f"TabDDPM source not mounted: {tabddpm_root}" +env = os.environ.copy() +env["PYTHONPATH"] = tabddpm_root + (os.pathsep + env.get("PYTHONPATH", "")) + +# Reuse the compat wrapper (patches collections.Sequence for skorch) +wrapper = os.path.join(tabddpm_root, "_compat_run.py") +if not os.path.exists(wrapper): + with open(wrapper, "w") as f: + f.write( + "import collections, collections.abc\n" + "for _a in ('Sequence','MutableSequence','MutableMapping','Mapping'," + "'MutableSet','Set','Callable','Iterable','Iterator'):\n" + " if not hasattr(collections, _a): setattr(collections, _a, getattr(collections.abc, _a, None))\n" + "import sys, runpy\n" + "sys.argv = sys.argv[1:]\n" + "runpy.run_path(sys.argv[0], run_name='__main__')\n" + ) + +print(f"[TabDDPM] Sampling 15215 rows") +ret = subprocess.run( + [sys.executable, wrapper, "scripts/pipeline.py", + "--config", "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_074742.toml", + "--sample"], + cwd=tabddpm_root, + env=env +) +if ret.returncode != 0: + sys.exit(ret.returncode) + +# 将 .npy 输出转为 CSV（npy 在 TabDDPM 的 parent_dir，即 npy_dir） +info_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/info.json" +with open(info_path) as f: + info = json.load(f) + +output_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/output" +col_names = info.get("column_names", []) + +parts = [] +x_num_path = os.path.join(output_dir, "X_num_train.npy") +x_cat_path = os.path.join(output_dir, "X_cat_train.npy") +y_path = os.path.join(output_dir, "y_train.npy") + +if os.path.exists(x_num_path): + parts.append(np.load(x_num_path, allow_pickle=True)) +if os.path.exists(x_cat_path): + parts.append(np.load(x_cat_path, allow_pickle=True).astype(float)) +if os.path.exists(y_path): + y = np.load(y_path, allow_pickle=True) + parts.append(y.reshape(-1, 1) if y.ndim == 1 else y) + +if parts: + combined = np.concatenate(parts, axis=1) + if col_names and len(col_names) == combined.shape[1]: + df = pd.DataFrame(combined, columns=col_names) + else: + df = pd.DataFrame(combined) + df.to_csv("/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/tabddpm-n11-15215-20260425_074742.csv", index=False) + print(f"[TabDDPM] Saved {len(df)} rows -> /work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/tabddpm-n11-15215-20260425_074742.csv") +else: + print("[TabDDPM] WARNING: No output .npy files found") + sys.exit(1) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_train.py new file mode 100644 index 0000000000000000000000000000000000000000..fdcfa29ed835cba24003ca3ea301b1e117b10e28 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_train.py @@ -0,0 +1,32 @@ +import os, sys, subprocess + +tabddpm_root = "/workspace/tabddpm/code" +assert os.path.isdir(tabddpm_root), f"TabDDPM source not mounted: {tabddpm_root}" +env = os.environ.copy() +env["PYTHONPATH"] = tabddpm_root + (os.pathsep + env.get("PYTHONPATH", "")) + +# Write a wrapper that patches collections.Sequence (removed in Python 3.10+) +# before running pipeline.py - needed because skorch uses old API +wrapper = os.path.join(tabddpm_root, "_compat_run.py") +with open(wrapper, "w") as f: + f.write( + "import collections, collections.abc\n" + "for _a in ('Sequence','MutableSequence','MutableMapping','Mapping'," + "'MutableSet','Set','Callable','Iterable','Iterator'):\n" + " if not hasattr(collections, _a): setattr(collections, _a, getattr(collections.abc, _a, None))\n" + "import sys, runpy\n" + "sys.argv = sys.argv[1:]\n" + "runpy.run_path(sys.argv[0], run_name='__main__')\n" + ) + +print(f"[TabDDPM] Training, config=/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config.toml") +ret = subprocess.run( + [sys.executable, wrapper, "scripts/pipeline.py", + "--config", "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config.toml", + "--train"], + cwd=tabddpm_root, + env=env +) +if ret.returncode != 0: + sys.exit(ret.returncode) +print("[TabDDPM] Training complete") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config.toml b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config.toml new file mode 100644 index 0000000000000000000000000000000000000000..5254aa929707bd699360fd6e177ac9a238ff2fae --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config.toml @@ -0,0 +1,39 @@ +seed = 0 +parent_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/output" +real_data_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data" +model_type = "mlp" +num_numerical_features = 10 +device = "cuda:0" + +[model_params] +d_in = 10 +num_classes = 2 +is_y_cond = true + +[model_params.rtdl_params] +d_layers = [256, 256] +dropout = 0.0 + +[diffusion_params] +num_timesteps = 1000 +gaussian_loss_type = "mse" + +[train.main] +steps = 5000 +lr = 0.001 +weight_decay = 0.0 +batch_size = 256 + +[train.T] +seed = 0 +normalization = "quantile" +num_nan_policy = "__none__" +cat_nan_policy = "__none__" +cat_min_frequency = "__none__" +cat_encoding = "__none__" +y_policy = "default" + +[sample] +num_samples = 1000 +batch_size = 1000 +seed = 0 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260424_210918.toml b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260424_210918.toml new file mode 100644 index 0000000000000000000000000000000000000000..dd636aad3016567ed82ad431d3c667f4c964491b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260424_210918.toml @@ -0,0 +1,39 @@ +seed = 0 +parent_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/output" +real_data_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data" +model_type = "mlp" +num_numerical_features = 10 +device = "cuda:0" + +[model_params] +d_in = 10 +num_classes = 2 +is_y_cond = true + +[model_params.rtdl_params] +d_layers = [256, 256] +dropout = 0.0 + +[diffusion_params] +num_timesteps = 1000 +gaussian_loss_type = "mse" + +[train.main] +steps = 5000 +lr = 0.001 +weight_decay = 0.0 +batch_size = 256 + +[train.T] +seed = 0 +normalization = "quantile" +num_nan_policy = "__none__" +cat_nan_policy = "__none__" +cat_min_frequency = "__none__" +cat_encoding = "__none__" +y_policy = "default" + +[sample] +num_samples = 15215 +batch_size = 1000 +seed = 0 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_032428.toml b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_032428.toml new file mode 100644 index 0000000000000000000000000000000000000000..dd636aad3016567ed82ad431d3c667f4c964491b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_032428.toml @@ -0,0 +1,39 @@ +seed = 0 +parent_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/output" +real_data_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data" +model_type = "mlp" +num_numerical_features = 10 +device = "cuda:0" + +[model_params] +d_in = 10 +num_classes = 2 +is_y_cond = true + +[model_params.rtdl_params] +d_layers = [256, 256] +dropout = 0.0 + +[diffusion_params] +num_timesteps = 1000 +gaussian_loss_type = "mse" + +[train.main] +steps = 5000 +lr = 0.001 +weight_decay = 0.0 +batch_size = 256 + +[train.T] +seed = 0 +normalization = "quantile" +num_nan_policy = "__none__" +cat_nan_policy = "__none__" +cat_min_frequency = "__none__" +cat_encoding = "__none__" +y_policy = "default" + +[sample] +num_samples = 15215 +batch_size = 1000 +seed = 0 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_074742.toml b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_074742.toml new file mode 100644 index 0000000000000000000000000000000000000000..dd636aad3016567ed82ad431d3c667f4c964491b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config_sample_20260425_074742.toml @@ -0,0 +1,39 @@ +seed = 0 +parent_dir = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/output" +real_data_path = "/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data" +model_type = "mlp" +num_numerical_features = 10 +device = "cuda:0" + +[model_params] +d_in = 10 +num_classes = 2 +is_y_cond = true + +[model_params.rtdl_params] +d_layers = [256, 256] +dropout = 0.0 + +[diffusion_params] +num_timesteps = 1000 +gaussian_loss_type = "mse" + +[train.main] +steps = 5000 +lr = 0.001 +weight_decay = 0.0 +batch_size = 256 + +[train.T] +seed = 0 +normalization = "quantile" +num_nan_policy = "__none__" +cat_nan_policy = "__none__" +cat_min_frequency = "__none__" +cat_encoding = "__none__" +y_policy = "default" + +[sample] +num_samples = 15215 +batch_size = 1000 +seed = 0 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/X_num_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/X_num_test.npy new file mode 100644 index 0000000000000000000000000000000000000000..45e382a5c153e321e0c0540a987d576398fc6880 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/X_num_test.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/X_num_val.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/X_num_val.npy new file mode 100644 index 0000000000000000000000000000000000000000..cf45a16c7ac7b3bcd6ab4fbee0a9c4a698d53874 Binary files /dev/null and b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/X_num_val.npy differ diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/info.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/info.json new file mode 100644 index 0000000000000000000000000000000000000000..ad93acde945d46916d99f9a52fb19bb740a3d107 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/info.json @@ -0,0 +1,37 @@ +{ + "name": "benchmark_dataset", + "task_type": "multiclass", + "n_num_features": 10, + "n_cat_features": 0, + "train_size": 15215, + "num_col_idx": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9 + ], + "cat_col_idx": [], + "target_col_idx": [ + 10 + ], + "column_names": [ + "28.7967", + "16.0021", + "2.6449", + "0.3918", + "0.1982", + "27.7004", + "22.011", + "-8.2027", + "40.092", + "81.8828", + "g" + ], + "num_classes": 2 +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/y_test.npy b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/data/y_test.npy new file mode 100644 index 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0000000000000000000000000000000000000000..95ac5613b9eae0723cec9b0c38cc276231efea63 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/gen_20260321_161532.log @@ -0,0 +1,1008 @@ +/opt/conda/lib/python3.11/site-packages/skorch/__init__.py:6: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81. + import pkg_resources +mlp +Sample timestep 999 +Sample timestep 998 +Sample timestep 997 +Sample timestep 996 +Sample timestep 995 +Sample timestep 994 +Sample timestep 993 +Sample timestep 992 +Sample timestep 991 +Sample timestep 990 +Sample timestep 989 +Sample timestep 988 +Sample timestep 987 +Sample timestep 986 +Sample timestep 985 +Sample timestep 984 +Sample timestep 983 +Sample timestep 982 +Sample timestep 981 +Sample timestep 980 +Sample timestep 979 +Sample timestep 978 +Sample timestep 977 +Sample timestep 976 +Sample timestep 975 +Sample timestep 974 +Sample timestep 973 +Sample timestep 972 +Sample timestep 971 +Sample timestep 970 +Sample timestep 969 +Sample timestep 968 +Sample timestep 967 +Sample timestep 966 +Sample timestep 965 +Sample timestep 964 +Sample timestep 963 +Sample timestep 962 +Sample timestep 961 +Sample timestep 960 +Sample timestep 959 +Sample timestep 958 +Sample timestep 957 +Sample 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"/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/_tabddpm_sample.py", line 38, in + with open(info_path) as f: + ^^^^^^^^^^^^^^^ +FileNotFoundError: [Errno 2] No such file or directory: '/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/output/data/info.json' +[TabDDPM] Sampling 15215 rows diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/gen_20260425_032428.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/gen_20260425_032428.log new file mode 100644 index 0000000000000000000000000000000000000000..5dd200c6017b2f87b14a8d0ff58f8e460e3bc419 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/gen_20260425_032428.log @@ -0,0 +1,16006 @@ +mlp +Sample timestep 999 +Sample timestep 998 +Sample timestep 997 +Sample timestep 996 +Sample timestep 995 +Sample timestep 994 +Sample timestep 993 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timestep 28 +Sample timestep 27 +Sample timestep 26 +Sample timestep 25 +Sample timestep 24 +Sample timestep 23 +Sample timestep 22 +Sample timestep 21 +Sample timestep 20 +Sample timestep 19 +Sample timestep 18 +Sample timestep 17 +Sample timestep 16 +Sample timestep 15 +Sample timestep 14 +Sample timestep 13 +Sample timestep 12 +Sample timestep 11 +Sample timestep 10 +Sample timestep 9 +Sample timestep 8 +Sample timestep 7 +Sample timestep 6 +Sample timestep 5 +Sample timestep 4 +Sample timestep 3 +Sample timestep 2 +Sample timestep 1 +Sample timestep 0 +Discrete cols: [] +Num shape: (15215, 10) +Elapsed time: 0:00:33 +[TabDDPM] Sampling 15215 rows +[TabDDPM] Saved 15215 rows -> /work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/tabddpm-n11-15215-20260425_074742.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/input_snapshot.json 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"/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/train_20260321_161342.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/train_20260321_161342.log new file mode 100644 index 0000000000000000000000000000000000000000..e48a225096fa19b3788e82e0fddf513772ba0883 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/train_20260321_161342.log @@ -0,0 +1,19 @@ +/opt/conda/lib/python3.11/site-packages/skorch/__init__.py:6: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81. + import pkg_resources +[0] +10 +{'d_in': 10, 'num_classes': 2, 'is_y_cond': True, 'rtdl_params': {'d_layers': [256, 256], 'dropout': 0.0}} +mlp +Step 500/5000 MLoss: 0.0 GLoss: 0.3733 Sum: 0.3733 +Step 1000/5000 MLoss: 0.0 GLoss: 0.3687 Sum: 0.3687 +Step 1500/5000 MLoss: 0.0 GLoss: 0.3566 Sum: 0.3566 +Step 2000/5000 MLoss: 0.0 GLoss: 0.3527 Sum: 0.3527 +Step 2500/5000 MLoss: 0.0 GLoss: 0.3576 Sum: 0.3576 +Step 3000/5000 MLoss: 0.0 GLoss: 0.3544 Sum: 0.3544 +Step 3500/5000 MLoss: 0.0 GLoss: 0.3537 Sum: 0.3537 +Step 4000/5000 MLoss: 0.0 GLoss: 0.3499 Sum: 0.3499 +Step 4500/5000 MLoss: 0.0 GLoss: 0.3483 Sum: 0.3483 +Step 5000/5000 MLoss: 0.0 GLoss: 0.3513 Sum: 0.3513 +Elapsed time: 0:01:36 +[TabDDPM] Training, config=/work/output-SpecializedModels/n11/tabddpm/tabddpm-n11-20260321_161341/config.toml +[TabDDPM] Training complete diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/_tabpfgen_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/_tabpfgen_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..fcec0744fd898bd20fe0e156a7756c63d82af86a --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/_tabpfgen_generate.py @@ -0,0 +1,68 @@ +import numpy as np +import pandas as pd +import json +from tabpfgen import TabPFGen + +df = pd.read_csv("/work/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/staged/public/train.csv") +target_col = "g" + +feature_cols = [c for c in df.columns if c != target_col] + +# --- Label-encode categorical / object columns --- +cat_encodings = {} # col -> list of unique values (index = code) +for col in feature_cols: + if df[col].dtype == object or str(df[col].dtype) == 'category': + cats = sorted(df[col].dropna().unique().tolist(), key=str) + cat_map = {v: i for i, v in enumerate(cats)} + df[col] = df[col].map(cat_map).astype(float) + cat_encodings[col] = cats + print(f"[TabPFGen] Label-encoded '{col}' ({len(cats)} categories)") + +# Encode target if categorical +target_cats = None +if df[target_col].dtype == object or str(df[target_col].dtype) == 'category': + cats = sorted(df[target_col].dropna().unique().tolist(), key=str) + t_map = {v: i for i, v in enumerate(cats)} + df[target_col] = df[target_col].map(t_map).astype(float) + target_cats = cats + print(f"[TabPFGen] Label-encoded target '{target_col}' ({len(cats)} categories)") + +X = df[feature_cols].values.astype(np.float32) +y = df[target_col].values + +# Handle NaN +for i in range(X.shape[1]): + col_vals = X[:, i] + mask = np.isnan(col_vals) + if mask.any(): + mean_val = np.nanmean(col_vals) + X[mask, i] = mean_val if not np.isnan(mean_val) else 0.0 + +gen = TabPFGen( + n_sgld_steps=1000, + sgld_step_size=0.01, + sgld_noise_scale=0.01, + device="auto", +) + +print(f"[TabPFGen] Generating 15215 rows via generate_classification") +X_syn, y_syn = gen.generate_classification(X, y, n_samples=15215) + +syn_df = pd.DataFrame(X_syn, columns=feature_cols) +syn_df[target_col] = y_syn + +# --- Inverse label-encoding for categorical columns --- +for col, cats in cat_encodings.items(): + # Round to nearest integer index, clamp to valid range + codes = np.round(syn_df[col].values).astype(int) + codes = np.clip(codes, 0, len(cats) - 1) + syn_df[col] = [cats[c] for c in codes] + +if target_cats is not None: + codes = np.round(syn_df[target_col].values).astype(int) + codes = np.clip(codes, 0, len(target_cats) - 1) + syn_df[target_col] = [target_cats[c] for c in codes] + +syn_df = syn_df[list(df.columns)] +syn_df.to_csv("/work/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/tabpfgen-n11-15215-20260422_070321.csv", index=False) +print(f"[TabPFGen] Saved {len(syn_df)} rows -> /work/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/tabpfgen-n11-15215-20260422_070321.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/gen_20260422_070321.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/gen_20260422_070321.log new file mode 100644 index 0000000000000000000000000000000000000000..40dfe82806fd9a701231a8a08d4c539e34da60c6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/gen_20260422_070321.log @@ -0,0 +1,14 @@ +[TabPFGen] Label-encoded target 'g' (2 categories) +[TabPFGen] Generating 15215 rows via generate_classification +Step 0/1000 +Step 100/1000 +Step 200/1000 +Step 300/1000 +Step 400/1000 +Step 500/1000 +Step 600/1000 +Step 700/1000 +Step 800/1000 +Step 900/1000 +[TabPFGen] Saved 15214 rows -> /work/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/tabpfgen-n11-15215-20260422_070321.csv +[W421 23:08:04.989132953 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..9d61d688ac9e7a2c5d3965067fdfc236327a6d3c --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n11", + "model": "tabpfgen", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-train.csv", + "exists": true, + "size": 1197106, + "sha256": 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b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/staged/tabpfgen/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..f0d747bc5def37cfdb669038435bde8d0aec3be6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_183752/staged/tabpfgen/model_input_manifest.json @@ -0,0 +1,242 @@ +{ + "dataset_id": "n11", + "model": "tabpfgen", + "target_column": "g", + "task_type": "classification", + "column_schema": [ + { + "name": "28.7967", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14962, + "unique_ratio": 0.983372, + "example_values": [ + "33.9347", + "118.601", + "19.83", + "33.2691", + "13.1404" + ] + } + }, + { + "name": "16.0021", + "role": "feature", + "semantic_type": "numeric", + 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"/data/jialinzhang/SynthesizePipeline-server/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/temp/tabpfgen_regen_parallel_deadline/20260422_070318/n11/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/_tabpfgen_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/_tabpfgen_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..888afbad538067e7e30e1109ba13e5980bb1ea7f --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/_tabpfgen_generate.py @@ -0,0 +1,87 @@ +import numpy as np +import pandas as pd +import json +from tabpfgen import TabPFGen + +df = pd.read_csv("/work/temp/tabpfgen_regen_parallel_deadline/20260422_191739/n11/staged/public/train.csv") +target_col = "g" + +feature_cols = [c for c in df.columns if c != target_col] + +# --- Label-encode categorical / object columns --- +cat_encodings = {} # col -> list of unique values (index = code) +for col in feature_cols: + if df[col].dtype == object or str(df[col].dtype) == 'category': + cats = sorted(df[col].dropna().unique().tolist(), key=str) + cat_map = {v: i for i, v in enumerate(cats)} + df[col] = df[col].map(cat_map).astype(float) + cat_encodings[col] = cats + print(f"[TabPFGen] Label-encoded '{col}' ({len(cats)} categories)") + +# Encode target if categorical +target_cats = None +if df[target_col].dtype == object or str(df[target_col].dtype) == 'category': + cats = sorted(df[target_col].dropna().unique().tolist(), key=str) + t_map = {v: i for i, v in enumerate(cats)} + df[target_col] = df[target_col].map(t_map).astype(float) + target_cats = cats + print(f"[TabPFGen] Label-encoded target '{target_col}' ({len(cats)} categories)") + +X = df[feature_cols].values.astype(np.float32) +y = df[target_col].values +target_n = int(15215) + +# Handle NaN +for i in range(X.shape[1]): + col_vals = X[:, i] + mask = np.isnan(col_vals) + if mask.any(): + mean_val = np.nanmean(col_vals) + X[mask, i] = mean_val if not np.isnan(mean_val) else 0.0 + +gen = TabPFGen( + n_sgld_steps=1000, + sgld_step_size=0.01, + sgld_noise_scale=0.01, + device="auto", +) + +print(f"[TabPFGen] Generating {target_n} rows via generate_classification") +X_syn, y_syn = gen.generate_classification(X, y, n_samples=target_n) + +syn_df = pd.DataFrame(X_syn, columns=feature_cols) +syn_df[target_col] = y_syn + +# --- Inverse label-encoding for categorical columns --- +for col, cats in cat_encodings.items(): + # Round to nearest integer index, clamp to valid range + codes = np.round(syn_df[col].values).astype(int) + codes = np.clip(codes, 0, len(cats) - 1) + syn_df[col] = [cats[c] for c in codes] + +if target_cats is not None: + codes = np.round(syn_df[target_col].values).astype(int) + codes = np.clip(codes, 0, len(target_cats) - 1) + syn_df[target_col] = [target_cats[c] for c in codes] + +# Ensure output row count is strictly aligned with target_n. +if len(syn_df) > target_n: + print(f"[TabPFGen] Trimming rows: {len(syn_df)} -> {target_n}") + syn_df = syn_df.iloc[:target_n].copy() +elif len(syn_df) < target_n: + deficit = target_n - len(syn_df) + print(f"[TabPFGen] Padding rows: {len(syn_df)} -> {target_n} (deficit={deficit})") + if len(syn_df) > 0: + extra = syn_df.sample(n=deficit, replace=True, random_state=42) + syn_df = pd.concat([syn_df.reset_index(drop=True), extra.reset_index(drop=True)], ignore_index=True) + else: + # Defensive fallback: if generator returns empty, bootstrap from training rows. + syn_df = df[feature_cols + [target_col]].sample( + n=target_n, replace=True, random_state=42 + ).reset_index(drop=True) + +syn_df = syn_df[list(df.columns)] +if len(syn_df) != target_n: + raise RuntimeError(f"[TabPFGen] Row alignment failed: got {len(syn_df)}, expected {target_n}") +syn_df.to_csv("/work/temp/tabpfgen_regen_parallel_deadline/20260422_191739/n11/tabpfgen-n11-15215-20260422_191741.csv", index=False) +print(f"[TabPFGen] Saved {len(syn_df)} rows -> /work/temp/tabpfgen_regen_parallel_deadline/20260422_191739/n11/tabpfgen-n11-15215-20260422_191741.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/gen_20260422_191741.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/gen_20260422_191741.log new file mode 100644 index 0000000000000000000000000000000000000000..bfd7c32614fe5ae8c0a739414fb70c050169c5cd --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/gen_20260422_191741.log @@ -0,0 +1,15 @@ +[TabPFGen] Label-encoded target 'g' (2 categories) +[TabPFGen] Generating 15215 rows via generate_classification +Step 0/1000 +Step 100/1000 +Step 200/1000 +Step 300/1000 +Step 400/1000 +Step 500/1000 +Step 600/1000 +Step 700/1000 +Step 800/1000 +Step 900/1000 +[TabPFGen] Padding rows: 15214 -> 15215 (deficit=1) +[TabPFGen] Saved 15215 rows -> /work/temp/tabpfgen_regen_parallel_deadline/20260422_191739/n11/tabpfgen-n11-15215-20260422_191741.csv +[W422 11:21:07.953130620 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..9d61d688ac9e7a2c5d3965067fdfc236327a6d3c --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n11", + "model": "tabpfgen", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-train.csv", + "exists": true, + "size": 1197106, + "sha256": "f8253c0573a8f028c0b1e17752d214fbad956d0ace350d0dee3947f3dfafbc75" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-val.csv", + "exists": true, + "size": 149602, + "sha256": "7bde20178eb7de9523d4df719a3b461656f5704e3a03187c8289b2299dc8e458" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-test.csv", + "exists": true, + "size": 149859, + "sha256": "855b55969de2da1341d428edef6dedfdf0c60f016b9fa08efaf8a883af6a2c53" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_profile.json", + "exists": true, + "size": 5104, + "sha256": "0bd368d22598246defd33bda12c9872165f26e798cc462ec472d193f322fba07" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_contract_v1.json", + "exists": true, + "size": 5874, + "sha256": "ad3f01eafe9fb37754eed9be9bd058390fd3a4f54d8be4ceb4b5326a9901da9d" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..96904695eb041844b615aa285fda674f1da87108 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11-migrated-20260422_193053/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,235 @@ +{ + "dataset_id": "n11", + "target_column": "g", + "task_type": "classification", + "columns": [ + { + "name": "28.7967", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14962, + "unique_ratio": 0.983372, + "example_values": [ + "33.9347", + "118.601", + "19.83", + "33.2691", + "13.1404" + ] + } + }, + { + "name": "16.0021", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14662, + "unique_ratio": 0.963654, + "example_values": [ + "10.8862", + "44.2479", + "7.3303", + "9.1269", + "10.8527" + ] + } + }, + { 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b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/_tabpfgen_generate.py @@ -0,0 +1,68 @@ +import numpy as np +import pandas as pd +import json +from tabpfgen import TabPFGen + +df = pd.read_csv("/work/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/staged/public/train.csv") +target_col = "g" + +feature_cols = [c for c in df.columns if c != target_col] + +# --- Label-encode categorical / object columns --- +cat_encodings = {} # col -> list of unique values (index = code) +for col in feature_cols: + if df[col].dtype == object or str(df[col].dtype) == 'category': + cats = sorted(df[col].dropna().unique().tolist(), key=str) + cat_map = {v: i for i, v in enumerate(cats)} + df[col] = df[col].map(cat_map).astype(float) + cat_encodings[col] = cats + print(f"[TabPFGen] Label-encoded '{col}' ({len(cats)} categories)") + +# Encode target if categorical +target_cats = None +if df[target_col].dtype == object or str(df[target_col].dtype) == 'category': + cats = sorted(df[target_col].dropna().unique().tolist(), key=str) + t_map = {v: i for i, v in enumerate(cats)} + df[target_col] = df[target_col].map(t_map).astype(float) + target_cats = cats + print(f"[TabPFGen] Label-encoded target '{target_col}' ({len(cats)} categories)") + +X = df[feature_cols].values.astype(np.float32) +y = df[target_col].values + +# Handle NaN +for i in range(X.shape[1]): + col_vals = X[:, i] + mask = np.isnan(col_vals) + if mask.any(): + mean_val = np.nanmean(col_vals) + X[mask, i] = mean_val if not np.isnan(mean_val) else 0.0 + +gen = TabPFGen( + n_sgld_steps=1000, + sgld_step_size=0.01, + sgld_noise_scale=0.01, + device="auto", +) + +print(f"[TabPFGen] Generating 15215 rows via generate_classification") +X_syn, y_syn = gen.generate_classification(X, y, n_samples=15215) + +syn_df = pd.DataFrame(X_syn, columns=feature_cols) +syn_df[target_col] = y_syn + +# --- Inverse label-encoding for categorical columns --- +for col, cats in cat_encodings.items(): + # Round to nearest integer index, clamp to valid range + codes = np.round(syn_df[col].values).astype(int) + codes = np.clip(codes, 0, len(cats) - 1) + syn_df[col] = [cats[c] for c in codes] + +if target_cats is not None: + codes = np.round(syn_df[target_col].values).astype(int) + codes = np.clip(codes, 0, len(target_cats) - 1) + syn_df[target_col] = [target_cats[c] for c in codes] + +syn_df = syn_df[list(df.columns)] +syn_df.to_csv("/work/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/tabpfgen-n11-15215-20260420_233905.csv", index=False) +print(f"[TabPFGen] Saved {len(syn_df)} rows -> /work/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/tabpfgen-n11-15215-20260420_233905.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/gen_20260420_233905.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/gen_20260420_233905.log new file mode 100644 index 0000000000000000000000000000000000000000..ac9ad81b02c8e0b194aafa85c2d774512820395d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/gen_20260420_233905.log @@ -0,0 +1,14 @@ +[TabPFGen] Label-encoded target 'g' (2 categories) +[TabPFGen] Generating 15215 rows via generate_classification +Step 0/1000 +Step 100/1000 +Step 200/1000 +Step 300/1000 +Step 400/1000 +Step 500/1000 +Step 600/1000 +Step 700/1000 +Step 800/1000 +Step 900/1000 +[TabPFGen] Saved 15214 rows -> /work/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/tabpfgen-n11-15215-20260420_233905.csv +[W420 15:41:06.957922365 AllocatorConfig.cpp:28] Warning: PYTORCH_CUDA_ALLOC_CONF is deprecated, use PYTORCH_ALLOC_CONF instead (function operator()) diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..9d61d688ac9e7a2c5d3965067fdfc236327a6d3c --- /dev/null +++ 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a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/staged/tabpfgen/adapter_transforms_applied.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/staged/tabpfgen/adapter_transforms_applied.json new file mode 100644 index 0000000000000000000000000000000000000000..0637a088a01e8ddab3bf3fa98dbe804cbde1a0dc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/staged/tabpfgen/adapter_transforms_applied.json @@ -0,0 +1 @@ +[] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/staged/tabpfgen/model_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/staged/tabpfgen/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..b5a812e57b02dab8471aaa13d7d6bcff63e6d101 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/n11/staged/tabpfgen/model_input_manifest.json @@ -0,0 +1,242 @@ +{ + "dataset_id": "n11", + "model": "tabpfgen", + "target_column": "g", + "task_type": "classification", + "column_schema": [ + { + "name": "28.7967", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14962, + "unique_ratio": 0.983372, + "example_values": [ + "33.9347", + "118.601", + "19.83", + "33.2691", + "13.1404" + ] + } + }, + { + "name": "16.0021", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14662, + "unique_ratio": 0.963654, + "example_values": [ + "10.8862", + "44.2479", + "7.3303", + "9.1269", + "10.8527" + ] + } + }, + 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"example_values": [ + "0.2155", + "0.1003", + "0.4095", + "0.2553", + "0.4352" + ] + } + }, + { + "name": "27.7004", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14996, + "unique_ratio": 0.985606, + "example_values": [ + "-25.0369", + "-95.8266", + "2.8528", + "12.3981", + "14.3006" + ] + } + }, + { + "name": "22.011", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 14984, + "unique_ratio": 0.984818, + "example_values": [ + "-23.1659", + "14.9809", + "-9.7666", + "19.9081", + "-3.6418" + ] + } + }, + { + "name": "-8.2027", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + 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"semantic_type": "categorical", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "mode", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2, + "unique_ratio": 0.000131, + "example_values": [ + "g", + "h" + ] + } + } + ], + "public_manifest": "/data/jialinzhang/SynthesizePipeline-server/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/public_gate/staged_input_manifest.json", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/temp/nightly_runs/tabpfgen_generate_non_n_gpu1/n11/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/_tabpfgen_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/_tabpfgen_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..169cd44fb9183a898a0559aa60010c82ec0ab85e --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/_tabpfgen_generate.py @@ -0,0 +1,70 @@ +import numpy as np +import pandas as pd +import json +from tabpfgen import TabPFGen + +df = pd.read_csv("/work/DatasetNew/n11/n11-train.csv") +target_col = "g" +if target_col == "__LAST__": + target_col = df.columns[-1] + +feature_cols = [c for c in df.columns if c != target_col] + +# --- Label-encode categorical / object columns --- +cat_encodings = {} # col -> list of unique values (index = code) +for col in feature_cols: + if df[col].dtype == object or str(df[col].dtype) == 'category': + cats = sorted(df[col].dropna().unique().tolist(), key=str) + cat_map = {v: i for i, v in enumerate(cats)} + df[col] = df[col].map(cat_map).astype(float) + cat_encodings[col] = cats + print(f"[TabPFGen] Label-encoded '{col}' ({len(cats)} categories)") + +# Encode target if categorical +target_cats = None +if df[target_col].dtype == object or str(df[target_col].dtype) == 'category': + cats = sorted(df[target_col].dropna().unique().tolist(), key=str) + t_map = {v: i for i, v in enumerate(cats)} + df[target_col] = df[target_col].map(t_map).astype(float) + target_cats = cats + print(f"[TabPFGen] Label-encoded target '{target_col}' ({len(cats)} categories)") + +X = df[feature_cols].values.astype(np.float32) +y = df[target_col].values + +# Handle NaN +for i in range(X.shape[1]): + col_vals = X[:, i] + mask = np.isnan(col_vals) + if mask.any(): + mean_val = np.nanmean(col_vals) + X[mask, i] = mean_val if not np.isnan(mean_val) else 0.0 + +gen = TabPFGen( + n_sgld_steps=1000, + sgld_step_size=0.01, + sgld_noise_scale=0.01, + device="auto", +) + +print(f"[TabPFGen] Generating 15215 rows via generate_classification") +X_syn, y_syn = gen.generate_classification(X, y, n_samples=15215) + +syn_df = pd.DataFrame(X_syn, columns=feature_cols) +syn_df[target_col] = y_syn + +# --- Inverse label-encoding for categorical columns --- +for col, cats in cat_encodings.items(): + # Round to nearest integer index, clamp to valid range + codes = np.round(syn_df[col].values).astype(int) + codes = np.clip(codes, 0, len(cats) - 1) + syn_df[col] = [cats[c] for c in codes] + +if target_cats is not None: + codes = np.round(syn_df[target_col].values).astype(int) + codes = np.clip(codes, 0, len(target_cats) - 1) + syn_df[target_col] = [target_cats[c] for c in codes] + +syn_df = syn_df[list(df.columns)] +syn_df.to_csv("/work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen-n11-15215-20260318_060205.csv", index=False) +print(f"[TabPFGen] Saved {len(syn_df)} rows -> /work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen-n11-15215-20260318_060205.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/gen_20260318_060205.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/gen_20260318_060205.log new file mode 100644 index 0000000000000000000000000000000000000000..f90ad27c9f69aa60c8279d20fd3984f2939e697a --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/gen_20260318_060205.log @@ -0,0 +1,28 @@ + +========== +== CUDA == +========== + +CUDA Version 12.8.1 + +Container image Copyright (c) 2016-2023, NVIDIA CORPORATION & AFFILIATES. All rights reserved. + +This container image and its contents are governed by the NVIDIA Deep Learning Container License. +By pulling and using the container, you accept the terms and conditions of this license: +https://developer.nvidia.com/ngc/nvidia-deep-learning-container-license + +A copy of this license is made available in this container at /NGC-DL-CONTAINER-LICENSE for your convenience. + +[TabPFGen] Label-encoded target 'g' (2 categories) +[TabPFGen] Generating 15215 rows via generate_classification +Step 0/1000 +Step 100/1000 +Step 200/1000 +Step 300/1000 +Step 400/1000 +Step 500/1000 +Step 600/1000 +Step 700/1000 +Step 800/1000 +Step 900/1000 +[TabPFGen] Saved 15214 rows -> /work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen-n11-15215-20260318_060205.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen_meta.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen_meta.json new file mode 100644 index 0000000000000000000000000000000000000000..54b9efe05356809bd942c82780c92ceca15533c5 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen_meta.json @@ -0,0 +1,8 @@ +{ + "csv_path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-train.csv", + "json_path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_profile.json", + "target_col": "g", + "is_classification": true, + "n_rows": 15215, + "n_cols": 11 +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/train_20260318_060205.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/train_20260318_060205.log new file mode 100644 index 0000000000000000000000000000000000000000..cd979675de2d3e2903843d61903854df1dad566b --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/train_20260318_060205.log @@ -0,0 +1 @@ +[TabPFGen] No training needed (pretrained). Meta saved to /data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260318_060205/tabpfgen_meta.json diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/_tabpfgen_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/_tabpfgen_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..738ba706216d2402b225e564c9a097b48edbafe6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/_tabpfgen_generate.py @@ -0,0 +1,68 @@ +import numpy as np +import pandas as pd +import json +from tabpfgen import TabPFGen + +df = pd.read_csv("/work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/staged/public/train.csv") +target_col = "g" + +feature_cols = [c for c in df.columns if c != target_col] + +# --- Label-encode categorical / object columns --- +cat_encodings = {} # col -> list of unique values (index = code) +for col in feature_cols: + if df[col].dtype == object or str(df[col].dtype) == 'category': + cats = sorted(df[col].dropna().unique().tolist(), key=str) + cat_map = {v: i for i, v in enumerate(cats)} + df[col] = df[col].map(cat_map).astype(float) + cat_encodings[col] = cats + print(f"[TabPFGen] Label-encoded '{col}' ({len(cats)} categories)") + +# Encode target if categorical +target_cats = None +if df[target_col].dtype == object or str(df[target_col].dtype) == 'category': + cats = sorted(df[target_col].dropna().unique().tolist(), key=str) + t_map = {v: i for i, v in enumerate(cats)} + df[target_col] = df[target_col].map(t_map).astype(float) + target_cats = cats + print(f"[TabPFGen] Label-encoded target '{target_col}' ({len(cats)} categories)") + +X = df[feature_cols].values.astype(np.float32) +y = df[target_col].values + +# Handle NaN +for i in range(X.shape[1]): + col_vals = X[:, i] + mask = np.isnan(col_vals) + if mask.any(): + mean_val = np.nanmean(col_vals) + X[mask, i] = mean_val if not np.isnan(mean_val) else 0.0 + +gen = TabPFGen( + n_sgld_steps=1000, + sgld_step_size=0.01, + sgld_noise_scale=0.01, + device="auto", +) + +print(f"[TabPFGen] Generating 1000 rows via generate_classification") +X_syn, y_syn = gen.generate_classification(X, y, n_samples=1000) + +syn_df = pd.DataFrame(X_syn, columns=feature_cols) +syn_df[target_col] = y_syn + +# --- Inverse label-encoding for categorical columns --- +for col, cats in cat_encodings.items(): + # Round to nearest integer index, clamp to valid range + codes = np.round(syn_df[col].values).astype(int) + codes = np.clip(codes, 0, len(cats) - 1) + syn_df[col] = [cats[c] for c in codes] + +if target_cats is not None: + codes = np.round(syn_df[target_col].values).astype(int) + codes = np.clip(codes, 0, len(target_cats) - 1) + syn_df[target_col] = [target_cats[c] for c in codes] + +syn_df = syn_df[list(df.columns)] +syn_df.to_csv("/work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/tabpfgen-n11-1000-20260321_065352.csv", index=False) +print(f"[TabPFGen] Saved {len(syn_df)} rows -> /work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/tabpfgen-n11-1000-20260321_065352.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/gen_20260321_065352.log b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/gen_20260321_065352.log new file mode 100644 index 0000000000000000000000000000000000000000..ca9d73f33458a28b9a00b2ad7407e23284b58b8c --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/gen_20260321_065352.log @@ -0,0 +1,28 @@ + +========== +== CUDA == +========== + +CUDA Version 12.8.1 + +Container image Copyright (c) 2016-2023, NVIDIA CORPORATION & AFFILIATES. All rights reserved. + +This container image and its contents are governed by the NVIDIA Deep Learning Container License. +By pulling and using the container, you accept the terms and conditions of this license: +https://developer.nvidia.com/ngc/nvidia-deep-learning-container-license + +A copy of this license is made available in this container at /NGC-DL-CONTAINER-LICENSE for your convenience. + +[TabPFGen] Label-encoded target 'g' (2 categories) +[TabPFGen] Generating 1000 rows via generate_classification +Step 0/1000 +Step 100/1000 +Step 200/1000 +Step 300/1000 +Step 400/1000 +Step 500/1000 +Step 600/1000 +Step 700/1000 +Step 800/1000 +Step 900/1000 +[TabPFGen] Saved 1000 rows -> /work/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/tabpfgen-n11-1000-20260321_065352.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..9d61d688ac9e7a2c5d3965067fdfc236327a6d3c --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n11", + "model": "tabpfgen", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-train.csv", + "exists": true, + "size": 1197106, + "sha256": "f8253c0573a8f028c0b1e17752d214fbad956d0ace350d0dee3947f3dfafbc75" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-val.csv", + "exists": true, + "size": 149602, + "sha256": "7bde20178eb7de9523d4df719a3b461656f5704e3a03187c8289b2299dc8e458" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n11/n11-test.csv", + "exists": true, + "size": 149859, + "sha256": "855b55969de2da1341d428edef6dedfdf0c60f016b9fa08efaf8a883af6a2c53" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_profile.json", + "exists": true, + "size": 5104, + "sha256": "0bd368d22598246defd33bda12c9872165f26e798cc462ec472d193f322fba07" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n11/n11-dataset_contract_v1.json", + "exists": true, + "size": 5874, + "sha256": "ad3f01eafe9fb37754eed9be9bd058390fd3a4f54d8be4ceb4b5326a9901da9d" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tabpfgen/tabpfgen-n11-20260321_065351/public_gate/normalized_schema_snapshot.json new file mode 100644 index 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end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/staged/tvae/adapter_transforms_applied.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/staged/tvae/adapter_transforms_applied.json new file mode 100644 index 0000000000000000000000000000000000000000..0637a088a01e8ddab3bf3fa98dbe804cbde1a0dc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/staged/tvae/adapter_transforms_applied.json @@ -0,0 +1 @@ +[] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/staged/tvae/model_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/staged/tvae/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..b057754c53af4b9ce23e7a8e81b7d6a71db16808 --- /dev/null +++ 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"/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n11/tvae/tvae-n11-20260328_053538/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/_arf_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/_arf_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..b785366c566cf88650c92450ac3274cdd6983704 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/_arf_generate.py @@ -0,0 +1,6 @@ +import pickle +with open("/work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf_model.pkl", "rb") as f: + model = pickle.load(f) +syn = model.forge(n=7654) +syn.to_csv("/work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf-n20-7654-20260330_071030.csv", index=False) +print(f"[ARF] Generated 7654 rows -> /work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf-n20-7654-20260330_071030.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/_arf_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/_arf_train.py new file mode 100644 index 0000000000000000000000000000000000000000..107892e4592b0675ee93437c564860ca2133d128 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/_arf_train.py @@ -0,0 +1,19 @@ +import pickle +import pandas as pd +from arfpy import arf + +df = pd.read_csv("/work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/train.csv") +df = df.dropna(axis=1, how="all") +print(f"[ARF] Training on {len(df)} rows, {len(df.columns)} cols") + +model = arf.arf(x=df) +if hasattr(model, "fit"): + model.fit() +elif hasattr(model, "forde"): + model.forde() +else: + raise RuntimeError("arfpy API: no fit() / forde()") + +with open("/work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf_model.pkl", "wb") as f: + pickle.dump(model, f) +print(f"[ARF] Model saved -> /work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf_model.pkl") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/gen_20260328_032306.log b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/gen_20260328_032306.log new file mode 100644 index 0000000000000000000000000000000000000000..a85b578f1a7d9e15284f508b27ee24c4a83721df --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/gen_20260328_032306.log @@ -0,0 +1,3 @@ +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +[ARF] Generated 1000 rows -> /work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf-n20-1000-20260328_032306.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/gen_20260330_071030.log b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/gen_20260330_071030.log new file mode 100644 index 0000000000000000000000000000000000000000..5ba31ae7bb8d33495bcf6cceba899810a92f6331 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/gen_20260330_071030.log @@ -0,0 +1,3 @@ +/usr/local/lib/python3.10/site-packages/arfpy/arf.py:329: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]` + if self.factor_cols[j]: +[ARF] Generated 7654 rows -> /work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf-n20-7654-20260330_071030.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..b121b50da6528b9cc616e54895ea7f1c58bbf83d --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n20", + "model": "arf", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-train.csv", + "exists": true, + "size": 291758, + "sha256": "8768b9641a081ce32375b2739c6d0a6cdf66c1b03268c7db5f71ffd0019c97e2" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-val.csv", + "exists": true, + "size": 36657, + "sha256": "c2dae02f854856ba824310ecaca5c73a72a3302753396aa681c82b69d65b174c" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-test.csv", + "exists": true, + "size": 36232, + "sha256": "82c48522e82ea4c5e4c407ec9239e82558dcff1df413f3d0c382e77fa041bc88" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n20/n20-dataset_profile.json", + "exists": true, + "size": 2675, + "sha256": "7e7d747ab11beae1e58278dc8d6ae08d7ae6f14fe93f77a4fc1e6e589df21df4" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n20/n20-dataset_contract_v1.json", + "exists": true, + "size": 3047, + "sha256": "20f513ccbc9f09d5286bc8998a5772f42b78b40aa1ed60b079fa73edf9c78eec" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/normalized_schema_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/normalized_schema_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..8618a420d9d101a283cfb38feb336ac37687988f --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/normalized_schema_snapshot.json @@ -0,0 +1,112 @@ +{ + "dataset_id": "n20", + "target_column": "PE", + "task_type": "regression", + "columns": [ + { + "name": "AT", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2635, + "unique_ratio": 0.344264, + "example_values": [ + "29.64", + "10.63", + "27.09", + "24.2", + "15.31" + ] + } + }, + { + "name": "V", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 625, + "unique_ratio": 0.081657, + "example_values": [ + "67.790000000000006", + "37.5", + "59.15", + "57.85", + "52.75" + ] + } + }, + { + "name": "AP", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2365, + "unique_ratio": 0.308989, + "example_values": [ + "1009.99", + "1008.93", + "1013.02", + "1013.05", + "1025.21" + ] + } + }, + { + "name": "RH", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 4096, + "unique_ratio": 0.535145, + "example_values": [ + "51.23", + "98.14", + "55.18", + "70.290000000000006", + "55.22" + ] + } + }, + { + "name": "PE", + "role": "target", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 4351, + "unique_ratio": 0.568461, + "example_values": [ + "440.74", + "474.81", + "438.9", + "446.87", + "460.77" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/public_gate_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/public_gate_report.json new file mode 100644 index 0000000000000000000000000000000000000000..91364f3480d02e634daea210e8cae28ad5402b46 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/public_gate_report.json @@ -0,0 +1,37 @@ +{ + "dataset_id": "n20", + "status": "pass", + "checks": [ + { + "check_id": "PG001_csv_parse_ok", + "status": "pass" + }, + { + "check_id": "PG002_split_header_consistent", + "status": "pass" + }, + { + "check_id": "PG003_profile_header_match", + "status": "pass" + }, + { + "check_id": "PG004_missing_token_normalized", + "status": "pass" + }, + { + "check_id": "PG005_semantic_type_validated", + "status": "pass" + }, + { + "check_id": "PG006_target_defined_and_valid", + "status": "pass" + } + ], + "target_column": "PE", + "task_type": "regression", + "input_splits": { + "train": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-train.csv", + "val": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-val.csv", + "test": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-test.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/staged_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/staged_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..e018061f8b136471a9af0a539491b613471392a8 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/staged_input_manifest.json @@ -0,0 +1,117 @@ +{ + "dataset_id": "n20", + "target_column": "PE", + "task_type": "regression", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/public_gate_report.json", + "column_schema": [ + { + "name": "AT", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2635, + "unique_ratio": 0.344264, + "example_values": [ + "29.64", + "10.63", + "27.09", + "24.2", + "15.31" + ] + } + }, + { + "name": "V", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 625, + "unique_ratio": 0.081657, + "example_values": [ + "67.790000000000006", + "37.5", + "59.15", + "57.85", + "52.75" + ] + } + }, + { + "name": "AP", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2365, + "unique_ratio": 0.308989, + "example_values": [ + "1009.99", + "1008.93", + "1013.02", + "1013.05", + "1025.21" + ] + } + }, + { + "name": "RH", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 4096, + "unique_ratio": 0.535145, + "example_values": [ + "51.23", + "98.14", + "55.18", + "70.290000000000006", + "55.22" + ] + } + }, + { + "name": "PE", + "role": "target", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 4351, + "unique_ratio": 0.568461, + "example_values": [ + "440.74", + "474.81", + "438.9", + "446.87", + "460.77" + ] + } + } + ] +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..fe2598f643dccfb56397cac47d34ae8ffd032562 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/runtime_result.json @@ -0,0 +1,14 @@ +{ + "dataset_id": "n20", + "model": "arf", + "run_id": "arf-n20-20260328_032219", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "skipped", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf-n20-7654-20260330_071030.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/adapter_report.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/adapter_report.json new file mode 100644 index 0000000000000000000000000000000000000000..0642aa58501454659468296072b27c5b305194f4 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/adapter_report.json @@ -0,0 +1,7 @@ +{ + "adapter_ready_status": "pass", + "adapter_fail_reason_code": null, + "adapter_fail_detail": null, + "adapter_transforms_applied": [], + "model_input_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/model_input_manifest.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/adapter_transforms_applied.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/adapter_transforms_applied.json new file mode 100644 index 0000000000000000000000000000000000000000..0637a088a01e8ddab3bf3fa98dbe804cbde1a0dc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/adapter_transforms_applied.json @@ -0,0 +1 @@ +[] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/model_input_manifest.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/model_input_manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..fe989a5a331d6f38b91a45055d7ec962db3e8dcc --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/arf/model_input_manifest.json @@ -0,0 +1,119 @@ +{ + "dataset_id": "n20", + "model": "arf", + "target_column": "PE", + "task_type": "regression", + "column_schema": [ + { + "name": "AT", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2635, + "unique_ratio": 0.344264, + "example_values": [ + "29.64", + "10.63", + "27.09", + "24.2", + "15.31" + ] + } + }, + { + "name": "V", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 625, + "unique_ratio": 0.081657, + "example_values": [ + "67.790000000000006", + "37.5", + "59.15", + "57.85", + "52.75" + ] + } + }, + { + "name": "AP", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 2365, + "unique_ratio": 0.308989, + "example_values": [ + "1009.99", + "1008.93", + "1013.02", + "1013.05", + "1025.21" + ] + } + }, + { + "name": "RH", + "role": "feature", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 4096, + "unique_ratio": 0.535145, + "example_values": [ + "51.23", + "98.14", + "55.18", + "70.290000000000006", + "55.22" + ] + } + }, + { + "name": "PE", + "role": "target", + "semantic_type": "numeric", + "nullable": false, + "missing_tokens": [], + "parse_format": null, + "impute_strategy": "median", + "profile_stats": { + "missing_rate": 0.0, + "unique_count": 4351, + "unique_ratio": 0.568461, + "example_values": [ + "440.74", + "474.81", + "438.9", + "446.87", + "460.77" + ] + } + } + ], + "public_manifest": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/staged_input_manifest.json", + "train_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/train.csv", + "val_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/val.csv", + "test_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/test.csv", + "features_json": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/staged_features.json", + "public_gate_report": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/public_gate/public_gate_report.json" +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/staged_features.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/staged_features.json new file mode 100644 index 0000000000000000000000000000000000000000..9394440ffb1562950776e6fb8b6683125b5a8ec2 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/staged_features.json @@ -0,0 +1,27 @@ +[ + { + "feature_name": "AT", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "V", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "AP", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "RH", + "data_type": "continuous", + "is_target": false + }, + { + "feature_name": "PE", + "data_type": "continuous", + "is_target": true + } +] \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/test.csv b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/test.csv new file mode 100644 index 0000000000000000000000000000000000000000..bb807ac5b23b2e7dfb39b3a5378cd452e84d56ed --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/staged/public/test.csv @@ -0,0 +1,959 @@ +AT,V,AP,RH,PE +14.72,41.23,997.91,82.69,467.11 +11.94,40.66,1016.61,96.11,467.99 +27.56,66.93,1016.81,55.59,444.1 +9.59,34.69,1027.65,75.31999999999998,478.88 +7.46,41.82,1032.67,74.59,483.11 +15.52,36.99,1006.86,89.7,467.9 +15.47,43.13,1015.11,50.5,466.63 +21.84,45.09,1013.94,46.11,450.88 +19.83,51.43,1007.36,91.64,449.45 +29.3,67.69,1007.17,55.5,430.72 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+13.66,40.55,1005.46,96.02,467.7 diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/train_20260328_032219.log b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/train_20260328_032219.log new file mode 100644 index 0000000000000000000000000000000000000000..292abfe40019c8a1132f81bb546503057384230c --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/train_20260328_032219.log @@ -0,0 +1,4 @@ +[ARF] Training on 7654 rows, 5 cols +Initial accuracy is 0.9231774235693755 +Iteration number 1 reached accuracy of 0.4794878494904625. +[ARF] Model saved -> /work/output-SpecializedModels/n20/arf/arf-n20-20260328_032219/arf_model.pkl diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/_tvae_generate.py b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/_tvae_generate.py new file mode 100644 index 0000000000000000000000000000000000000000..334417c718a60d665c90113e00180ea498857996 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/_tvae_generate.py @@ -0,0 +1,21 @@ +import os, sys +sys.path.insert(0, "/work") +from src.SpecificModels.ctgan_rdt_inverse_fix import apply_ctgan_inverse_fix +apply_ctgan_inverse_fix() +import pandas as pd +from ctgan.synthesizers.tvae import TVAE +os.environ.setdefault("LOKY_MAX_CPU_COUNT", "8") +os.environ.setdefault("OPENBLAS_NUM_THREADS", "8") +os.environ.setdefault("MKL_NUM_THREADS", "8") +model = TVAE.load("/work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/models_300epochs/tvae_300epochs.pt") +total = 7654 +chunk = min(50000, total) if total > 50000 else total +parts = [] +left = total +while left > 0: + take = min(chunk, left) + parts.append(model.sample(take)) + left -= take +samples = pd.concat(parts, ignore_index=True) if len(parts) > 1 else parts[0] +samples.to_csv("/work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae-n20-7654-20260423_040345.csv", index=False) +print(f"[TVAE] Generated {total} rows (chunks={len(parts)}) -> /work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae-n20-7654-20260423_040345.csv") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/_tvae_train.py b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/_tvae_train.py new file mode 100644 index 0000000000000000000000000000000000000000..a683ee11aac45b9aa960fdccdff55451a2220a07 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/_tvae_train.py @@ -0,0 +1,16 @@ +import json, sys +import pandas as pd +from ctgan.data import read_csv +from ctgan.synthesizers.tvae import TVAE + +csv_path = "/work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/staged/public/train.csv" +meta_path = "/work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae_metadata.json" +save_path = "/work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/models_300epochs/tvae_300epochs.pt" +epochs = 300 + +data, discrete_columns = read_csv(csv_path, meta_path, header=True, discrete=None) +print(f"[TVAE] Training on {len(data)} rows, {len(data.columns)} cols, epochs={epochs}") +model = TVAE(epochs=epochs, batch_size=500) +model.fit(data, discrete_columns) +model.save(save_path) +print(f"[TVAE] Model saved -> {save_path}") diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260328_055255.log b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260328_055255.log new file mode 100644 index 0000000000000000000000000000000000000000..4929c027d9947acd0f7cc1ae903a0700904bf462 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260328_055255.log @@ -0,0 +1 @@ +[TVAE] Generated 1000 rows -> /work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae-n20-1000-20260328_055255.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260330_071012.log b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260330_071012.log new file mode 100644 index 0000000000000000000000000000000000000000..7d177c5b6c4bf04ab7a3501308c451ce43820046 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260330_071012.log @@ -0,0 +1 @@ +[TVAE] Generated 7654 rows -> /work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae-n20-7654-20260330_071012.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260423_040345.log b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260423_040345.log new file mode 100644 index 0000000000000000000000000000000000000000..fb64192119f33d919d96de5bf46deaa350f97482 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/gen_20260423_040345.log @@ -0,0 +1,4 @@ +/opt/conda/lib/python3.11/site-packages/sklearn/base.py:463: InconsistentVersionWarning: Trying to unpickle estimator BayesianGaussianMixture from version 1.7.2 when using version 1.8.0. This might lead to breaking code or invalid results. Use at your own risk. For more info please refer to: +https://scikit-learn.org/stable/model_persistence.html#security-maintainability-limitations + warnings.warn( +[TVAE] Generated 7654 rows (chunks=1) -> /work/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae-n20-7654-20260423_040345.csv diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/input_snapshot.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/input_snapshot.json new file mode 100644 index 0000000000000000000000000000000000000000..4f87e960764e5bbffe32e8b2f91fc5df5c018742 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/input_snapshot.json @@ -0,0 +1,36 @@ +{ + "dataset_id": "n20", + "model": "tvae", + "inputs": { + "train_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-train.csv", + "exists": true, + "size": 291758, + "sha256": "8768b9641a081ce32375b2739c6d0a6cdf66c1b03268c7db5f71ffd0019c97e2" + }, + "val_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-val.csv", + "exists": true, + "size": 36657, + "sha256": "c2dae02f854856ba824310ecaca5c73a72a3302753396aa681c82b69d65b174c" + }, + "test_csv": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/n20/n20-test.csv", + "exists": true, + "size": 36232, + "sha256": "82c48522e82ea4c5e4c407ec9239e82558dcff1df413f3d0c382e77fa041bc88" + }, + "profile_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n20/n20-dataset_profile.json", + "exists": true, + "size": 2675, + "sha256": "7e7d747ab11beae1e58278dc8d6ae08d7ae6f14fe93f77a4fc1e6e589df21df4" + }, + "contract_json": { + "path": "/data/jialinzhang/SynthesizePipeline-server/DatasetNew/artifacts/data_core/tabular/n20/n20-dataset_contract_v1.json", + "exists": true, + "size": 3047, + "sha256": "20f513ccbc9f09d5286bc8998a5772f42b78b40aa1ed60b079fa73edf9c78eec" + } + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/runtime_result.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/runtime_result.json new file mode 100644 index 0000000000000000000000000000000000000000..95f1db98c96fec6e88c8bde4fd2c13d2d86a51c6 --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/runtime_result.json @@ -0,0 +1,14 @@ +{ + "dataset_id": "n20", + "model": "tvae", + "run_id": "tvae-n20-20260328_054850", + "public_gate_status": "pass", + "adapter_ready_status": "pass", + "train_status": "skipped", + "generate_status": "success", + "reason_code": null, + "reason_detail": null, + "artifacts": { + "synthetic_csv": "/data/jialinzhang/SynthesizePipeline-server/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae-n20-7654-20260423_040345.csv" + } +} \ No newline at end of file diff --git a/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae_metadata.json b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae_metadata.json new file mode 100644 index 0000000000000000000000000000000000000000..9718f799216c76afaec80445613f5b1a930e899a --- /dev/null +++ b/SynthesizePipeline_Archive/output-SpecializedModels/n20/tvae/tvae-n20-20260328_054850/tvae_metadata.json @@ -0,0 +1,24 @@ +{ + "columns": [ + { + "name": "AT", + "type": "continuous" + }, + { + "name": "V", + "type": "continuous" + }, + { + "name": "AP", + "type": "continuous" + }, + { + "name": "RH", + "type": "continuous" + }, + { + "name": "PE", + "type": "continuous" + } + ] +} \ No newline at end of file